Measuring instruments, systems and magnetic gradiometers

ABSTRACT

Embodiments generally relate to a measuring instrument. The measuring instrument may comprise: a sensor to measure a property of the local environment; a mechanism configured to cause the sensor to move along a predetermined path relative to a fixed reference frame of the instrument; and a signal processing system configured to receive a processor sensor signal generated by the sensor, perform a Fourier transform on the sensor signal to identify frequency components of the sensor signal, and compare the frequency components of the sensor signal with frequency components associated with the predetermined path to determine a measurement of the property of the local environment. The mechanism may comprise a first member having a first axis and a second axis that is different from the first axis. The mechanism may be configured to cause the first member and the sensor to rotate about the first axis, and to cause the sensor to rotate about the second axis. The sensor may be spatially offset from the first axis. The sensor may be configured to measure one or more vector components of a local force field, such as a magnetic field, for example, and the measuring instrument may comprise a magnetometer or magnetic gradiometer, for example. The measuring instruments disclosed may be suited to downhole applications, such as in a measurement while drilling system, for example.

TECHNICAL FIELD

Embodiments generally relate to measuring instruments, systems andmethods, and in particular to magnetic gradiometers.

BACKGROUND

Various instruments, systems and methods are known for measuringphysical quantities. For example, magnetic gradiometers may be used formeasuring physical quantities such as magnetic fields, for example.

However, existing magnetic gradiometers have some drawbacks associatedwith them, particularly when used for certain applications such as in aborehole for use in mining, for example.

It is desired to address or ameliorate one or more problems,disadvantages or drawbacks of prior measuring instruments, systems ormethods, or to at least provide a useful alternative thereto.

Any discussion of documents, acts, materials, devices, articles or thelike which has been included in the present specification is not to betaken as an admission that any or all of these matters form part of theprior art base or were common general knowledge in the field relevant tothe present disclosure as it existed before the priority date of eachclaim of this application.

Throughout this specification the word “comprise”, or variations such as“comprises” or “comprising”, will be understood to imply the inclusionof a stated element, integer or step, or group of elements, integers orsteps, but not the exclusion of any other element, integer or step, orgroup of elements, integers or steps.

SUMMARY

Some embodiments relate to a measuring instrument comprising: a sensorto measure a property of the local environment; a mechanism configuredto cause the sensor to move along a predetermined path relative to afixed reference frame of the instrument; and a signal processing systemconfigured to receive a sensor signal generated by the sensor, perform aFourier transform on the sensor signal to identify frequency componentsof the sensor signal, and compare the frequency components of the sensorsignal with frequency components associated with the predetermined pathto determine a measurement of the property of the local environment;wherein the mechanism comprises a first member, wherein the first memberhas a first axis and a second axis that is different from the firstaxis, wherein the mechanism is configured to cause the first member andthe sensor to rotate about the first axis, and to cause the sensor torotate about the second axis, and wherein the sensor is spatially offsetfrom the first axis.

The sensor may be spatially offset from the second axis. The second axismay be spatially offset from the first axis. The second axis may beperpendicular to the first axis. The second axis may be inclined at anacute angle relative to the first axis.

There may be a first angular velocity associated with rotation of thesensor and the first member about the first axis and a second angularvelocity associated with rotation of the sensor about the second axis.Rotation of the sensor and the first member about the first axis may becoupled to rotation of the sensor about the second axis such that thefirst angular velocity is related to the second angular velocity by apredetermined ratio between the first and second angular velocities. Insome embodiments, the first and second angular velocities may bevariable with time. In some embodiments, the first and second angularvelocities may be constant.

In some embodiments, the mechanism may comprise a first sub-mechanism tocause the first member and the sensor to rotate about the first axis;and a second sub-mechanism to cause the sensor to rotate about thesecond axis. The second sub-mechanism may comprise the first member. Thefirst sub-mechanism may comprise a support to support the first member.The first member may comprises an elongate arm. The second sub-mechanismmay further comprise an arm extension to spatially offset the sensorfrom the second axis.

In some embodiments, the Fourier transform may be performed by a fastFourier transform chip in the signal processing system. The signalprocessing system may be further configured to perform a Fouriertransform on the sensor signal to identify frequency components of thesensor signal for comparison with frequency components associated withthe predetermined path. The signal processing system may furthercomprise a signal processor to receive the sensor signal and a computerprocessor to analyse and compare the frequency components of the sensorsignal with frequency components associated with the predetermined path.

The sensor may be configured to measure any one or more of: a vectorcomponent of a local force field, multiple vector components of a localforce field, and the field strength of a local force field. The forcefield may be a magnetic field, electric field, or gravitational field,for example. The measuring instrument may comprise a magneticgradiometer. The sensor may comprise a total magnetic intensitymagnetometer. The sensor may comprise a uniaxial magnetometer. Thesensor may comprise a fluxgate magnetometer.

In some embodiments, the sensor may be a first sensor and the measuringinstrument may further comprise a second sensor offset from the firstaxis on an opposite side of the first axis from the first sensor,wherein the mechanism is configured to cause the second sensor to rotateabout the first axis in the same direction and with the same angularvelocity as the first sensor and to cause the second sensor to rotateabout the second axis in the same direction and with the same angularvelocity as the first sensor. Rotation of the second sensor about thesecond axis may be out of phase relative to rotation of the first sensorabout the second axis. Rotation of the second sensor about the secondaxis may be about ninety degrees (90°) out of phase relative to rotationof the first sensor about the second axis.

In other embodiments, the measuring instrument may further comprise asecond member having a third axis which is different from the first andsecond axes, wherein the mechanism is configured to cause the secondmember to rotate about the first axis. The second member may be coupledto the first member or may be coupled to the mechanism. The sensor maybe a first sensor and the measuring instrument may further comprise asecond sensor, wherein the mechanism is configured to cause the secondsensor to rotate about the first axis in the same direction and with thesame angular velocity as rotation of the first sensor about the firstaxis and to cause the second sensor to rotate about the third axis inthe same direction and with the same angular velocity as rotation of thefirst sensor about the second axis.

Rotation of the second sensor about the first axis may be out of phaserelative to rotation of the first sensor about the first axis. Rotationof the second sensor about the first axis may be about one hundred andeighty degrees (180°) out of phase relative to rotation of the firstsensor about the first axis. Rotation of the second sensor about thethird axis may be out of phase relative to rotation of the first sensorabout the second axis. Rotation of the second sensor about the thirdaxis may be about ninety degrees (90°) out of phase relative to rotationof the first sensor about the second axis.

In some embodiments, the measuring instrument may further comprise athird sensor offset from the first axis and a fourth sensor offset fromthe first axis on an opposite side of the first axis from the thirdsensor, wherein the mechanism is configured to cause each of the thirdand fourth sensors to rotate about the first axis in the same directionand with the same angular velocity as rotation of the first and secondsensors about the first axis, and wherein the mechanism is configured tocause each of the third and fourth sensors to rotate about the thirdaxis in the same direction and with the same angular velocity asrotation of the first and second sensors about the second axis.

Rotation of the fourth sensor about the third axis may be out of phaserelative to rotation of the third sensor about the third axis. Rotationof the fourth sensor about the third axis may be about ninety degrees(90°) out of phase relative to rotation of the third sensor about thethird axis. Rotation of the third and fourth sensors about the firstaxis may be out of phase relative to rotation of the first and secondsensors about the first axis. Rotation of the third and fourth sensorsabout the third axis may be out of phase relative to rotation of thefirst and second sensors about the second axis.

The third axis may parallel to the second axis. The third axis may bespatially offset from the second axis. The third axis may be spatiallyoffset from the second axis in a direction parallel to the first axis.The third axis may be spatially offset from the first axis. The thirdaxis may be perpendicular to the first axis. The third axis may beinclined at an acute angle relative to the first axis.

In some embodiments, the signal processing system is configured toreceive all sensor signals generated by the sensors, perform a Fouriertransform on each of the sensor signals to identify frequency componentsof the sensor signals, and compare frequency components of the sensorsignals with frequency components associated with predetermined paths ofthe sensors to determine measurements of the property of the localenvironment. The Fourier transforms may be performed by one or more fastFourier transform chips in the signal processing system. The signalprocessing system may comprise a signal processor to receive the sensorsignals and a computer processor to analyse and compare the frequencycomponents of the sensor signals with frequency components associatedwith the predetermined paths of the sensors.

The sensors may be configured to measure any one or more of: a vectorcomponent of a local force field, multiple vector components of a localforce field, and the field strength of a local force field. the sensorscomprise total magnetic intensity magnetometers. The force field may bea magnetic field, electric field, or gravitational field, for example.The measuring instrument may comprise a magnetic gradiometer. Thesensors may comprise total magnetic intensity magnetometers. The sensorsmay comprise uniaxial magnetometers. The sensors may comprise fluxgatemagnetometers.

In some embodiments, the measuring instrument may further comprise anactuator coupled to and configured to drive the mechanism. The actuatormay comprise an electric motor or a turbine, for example. The measuringinstrument may further comprise a controller connected to the actuatorto control the actuator.

In some embodiments, the measuring instrument may further comprise oneor more angular position sensors connected to the signal processor toprovide angular position information for one or more components of themechanism.

The measuring instrument may further comprise a power source to poweroperation of the measuring instrument. The power source may power anyone or more of: the actuator or actuators, the controller, the sensor orsensors, the signal processing system, the signal processor, and thecomputer processor.

Some embodiments relate to a measuring instrument comprising a magneticgradiometer, the magnetic gradiometer comprising: a single uniaxialmagnetometer; a mechanism configured to move the magnetometer along apredetermined path relative to a fixed reference frame of the measuringinstrument; and a signal processing system configured to receive amagnetometer signal generated by the magnetometer, perform a Fouriertransform on the magnetometer signal to identify frequency components ofthe magnetometer signal, and compare frequency components of themagnetometer signal with frequency components associated with thepredetermined path to determine two or more independent components of amagnetic gradient tensor of the local magnetic field.

In some embodiments, the motion of the magnetometer along thepredetermined path may allow two, three, four or five independentcomponents of the magnetic gradient tensor to be determined by comparingfrequency components of the magnetometer signal with frequencycomponents associated with the predetermined path.

Some embodiments relate to a measuring instrument comprising a magneticgradiometer, the magnetic gradiometer comprising: a single totalmagnetic intensity magnetometer; a mechanism configured to move themagnetometer along a predetermined path relative to a fixed referenceframe of the measuring instrument; and a signal processing systemconfigured to receive a magnetometer signal generated by themagnetometer, perform a Fourier transform on the magnetometer signal toidentify frequency components of the magnetometer signal, and comparefrequency components of the magnetometer signal with frequencycomponents associated with the predetermined path to determine a totalmagnetic intensity and two or more components of a vector gradient ofthe total magnetic intensity of a local magnetic field.

The Fourier transform may be performed by a fast Fourier transform chipin the signal processing system. The signal processing system maycomprise a signal processor to receive the magnetometer signal and acomputer processor to analyse and compare frequency components of themagnetometer signal with frequency components associated with thepredetermined path.

In some embodiments, the measuring instrument may further comprise oneor more angular position sensors coupled to the mechanism to provide areference for comparing frequency components of the magnetometer signalwith frequency components associated with the predetermined path.

In some embodiments, the measuring instrument may further comprise oneor more further magnetometers to provide further data to assist indetermining one or more characteristics of the local magnetic field bycomparing frequency components of magnetometer signals from the one ormore further magnetometers with frequency components associated withpredetermined paths of the one or more further magnetometers.

In some embodiments, the single total magnetic intensity magnetometermay be a first total magnetic intensity magnetometer and the measuringinstrument may further comprise a second total magnetic intensitymagnetometer, wherein the predetermined path of the first magnetometeris a circular orbit, and wherein the mechanism is further configured tomove the second magnetometer along the predetermined path on an opposingside of the circular orbit to the first magnetometer.

Some embodiments relate to a magnetic detection system comprising ameasuring instrument according to any one of the described measuringinstruments. Some embodiments relate to a surveying system comprising ameasuring instrument according to any one of the described measuringinstruments. Some embodiments relate to a measurement while drillingsystem comprising a measuring instrument according to any one of thedescribed measuring instruments.

Some embodiments relate to a drilling system comprising a measuringinstrument according to any one of the described measuring instruments.The drilling system may further comprise: a drill bit for cutting aborehole in rock or earth; a drill string to drive the drill bit, thedrill string having a lumen to deliver drilling mud to the drill bit;and a drilling rig to drive rotation of the drill string and drill bit.

In some embodiments, at least part of the measuring instrument may bemounted to the drill string. At least part of the mechanism of themeasuring instrument may be driven by the drilling rig. At least part ofthe mechanism of the instrument may be driven by drilling mud pressurevia a turbine coupled to the mechanism.

In some embodiments, the first sub-mechanism may comprise the drillstring. And in some embodiments, the second sub-mechanism may comprise aturbine positioned within the lumen of the drill string to drive thesecond sub-mechanism.

In some other embodiments, the second sub-mechanism may comprise anelectric motor. The motor may be powered by a generator that is poweredby a drilling mud turbine, for example, or another suitable electricpower source.

Some embodiments relate to a method for simultaneously measuring twomagnetic field components and two magnetic gradient components using asingle uniaxial magnetic field sensor following a circular orbitrelative to a magnetic field.

Some embodiments relate to a method for simultaneously measuring threemagnetic field components and five independent components of thecomplete magnetic gradient tensor using two uniaxial magnetic fieldsensors following circular orbits relative to a magnetic field, the twoorbits being axially offset relative to an axis of rotation of theorbits.

Some embodiments relate to a method for simultaneously measuring thetotal magnetic intensity and two components of the vector gradient ofthe total magnetic intensity using a single scalar magnetic sensorfollowing a circular orbit relative to a magnetic field.

Some embodiments relate to a method for simultaneously measuring thetotal magnetic intensity and three components of the vector gradient ofthe total magnetic intensity using two scalar magnetic sensors followingcircular orbits relative to a magnetic field, the two orbits beingaxially offset relative to an axis of rotation of the orbits.

Some embodiments relate to a method for simultaneously measuring threemagnetic field components and four magnetic gradient components using asingle uniaxial magnetic field sensor that undergoes orbital motionwhile spinning about a different axis. The different axis may be alignedwith a radius of the orbital motion or may be aligned tangentially tothe orbital motion. In some embodiments, the different axis may beinclined relative to the radius, the tangent, or both the radius and thetangent of the orbital motion.

Some embodiments relate to a method for simultaneously measuring threemagnetic field components and the complete magnetic gradient tensorusing two uniaxial magnetic field sensors that undergo orbital motionwhile spinning about a different axis, as described above, the twoorbits being axially offset relative to an axis of rotation of theorbits.

Some embodiments relate to a method for simultaneously measuring threemagnetic field components and four magnetic gradient components usingdual uniaxial magnetic field sensors that undergo orbital motion whilespinning about a different axis, as described above. The dual sensorsfollowing the same orbital path while being located on opposite sides ofthe orbit.

Some embodiments relate to a method for simultaneously measuring threemagnetic field components and the complete magnetic gradient tensorusing two dual uniaxial magnetic field sensors, as described above.

Some embodiments relate to a method for simultaneously measuring threemagnetic field components and the complete magnetic gradient tensorusing dual uniaxial magnetic field sensors that undergo antipodalprecessional motion while spinning about a precessing axis.

Some embodiments relate to a method for simultaneously measuring threemagnetic field components and the complete magnetic gradient tensorusing a single uniaxial magnetic field sensor that undergoes orbitalmotion about a first axis while simultaneously orbiting a second axiswhich is orthogonal to the first axis.

BRIEF DESCRIPTION OF DRAWINGS

Embodiments are described in further detail, by way of example, withreference to the accompanying drawings, in which:

FIG. 1 is a block diagram of a measurement instrument according to someembodiments;

FIG. 2A is a perspective view of a sensor trajectory of a sensor movingwith respect to a first reference frame X of an instrument according tosome embodiments, the sensor orbits an origin of the first referenceframe X with a second reference frame X′;

FIG. 2B is a plan view of the trajectory of FIG. 2A;

FIG. 2C is a side view of the trajectory of FIG. 2A;

FIG. 2D is an orthogonal view of the x′z′-plane of the second referenceframe X′ of FIG. 2A showing the relative orientation of a sensitive axisu of the sensor;

FIG. 3A is a perspective view of a sensor trajectory of a sensor movingwith respect to a first reference frame X of an instrument according tosome embodiments, the sensor orbits an origin of the first referenceframe X with a second reference frame X′;

FIG. 3B is a plan view of the trajectory of FIG. 3A;

FIG. 3C is a side view of the trajectory of FIG. 3A;

FIG. 3D is a orthogonal view of the x′z′-plane of the second referenceframe X′ of FIG. 3A showing the rotation of a sensitive axis u of thesensor about the y′-axis;

FIG. 4A is a perspective view of a dual sensor configuration accordingto some embodiments, each sensor S1 and S2 following a similartrajectory to the trajectory of FIG. 3A;

FIG. 4B is a plan view of the configuration of FIG. 4A;

FIG. 4C is an orthogonal view of the x′z′-plane of the X′_(S1)-frame ofFIG. 4A showing the rotation of a sensitive axis u₁ of the sensor S1about the y′_(S1)-axis;

FIG. 4D is an orthogonal view of the x′z′-plane of the X′_(S2)-frame ofFIG. 4A showing the rotation of a sensitive axis u₂ of the sensor S2about the y′_(S2)-axis;

FIG. 5A is a perspective view of a sensor trajectory of a sensor movingwith respect to a first reference frame X of an instrument according tosome embodiments, the sensor orbits an origin of the first referenceframe X with a second reference frame X′;

FIG. 5B is a plan view of the trajectory of FIG. 5A;

FIG. 5C is a side view of the trajectory of FIG. 5A;

FIG. 5D is a orthogonal view of the z′y′-plane of the second referenceframe X′ of FIG. 5A showing the rotation of a sensitive axis u of thesensor about the x′-axis;

FIG. 6A is a perspective view of a dual sensor configuration accordingto some embodiments, each sensor S1 and S2 following a similartrajectory to the trajectory of FIG. 5A;

FIG. 6B is a plan view of the configuration of FIG. 6A;

FIG. 6C is an orthogonal view of the z′y′-plane of the X′_(S1)-frame ofFIG. 4A showing the rotation of a sensitive axis u₁ of the sensor S1about the x′_(S1)-axis;

FIG. 6D is an orthogonal view of the z′y′-plane of the X′_(S2)-frame ofFIG. 6A showing the rotation of a sensitive axis u₂ of the sensor S2about the x′_(S2)-axis;

FIG. 7A is a perspective view of a sensor trajectory of a sensor movingwith respect to a first reference frame X of an instrument according tosome embodiments, the sensor orbits an origin of the first referenceframe X with a second reference frame X′;

FIG. 7B is a plan view of the trajectory of FIG. 7A;

FIG. 7C is a side view of the trajectory of FIG. 7A;

FIG. 7D is an orthogonal view of the x′z′-plane of FIG. 7A showing therelative inclination of a third reference frame X″ with respect to thesecond reference frame X;

FIG. 7E is an orthogonal view of the x″y″-plane of the third referenceframe X″ of FIG. 7D showing the rotation of a sensitive axis u of thesensor about the z″-axis;

FIG. 8A is a perspective view of a sensor trajectory of a sensor movingwith respect to a first reference frame X of an instrument according tosome embodiments, the sensor orbits an origin of the first referenceframe X with a second reference frame X′;

FIG. 8B is a plan view of the trajectory of FIG. 8A;

FIG. 8C is a side view of the trajectory of FIG. 8A;

FIG. 8D is an orthogonal view of the z′y′-plane of FIG. 8A showing therelative inclination of a third reference frame X″ with respect to thesecond reference frame X;

FIG. 8E is an orthogonal view of the x″y″-plane of the third referenceframe X″ of FIG. 8D showing the rotation of a sensitive axis u of thesensor about the z″-axis;

FIG. 9A is a perspective view of a dual sensor configuration accordingto some embodiments, each sensor following a similar trajectory to thetrajectory of FIG. 8A with a third reference frame X″_(S1) of a firstsensor S1 and a third reference frame X″_(S2) of a second sensor S2;

FIG. 9B is a side view of the configuration of FIG. 9A;

FIG. 9C is an orthogonal view of the x″y″-plane of the X″_(S1)-frame ofFIG. 9A, looking along the z″_(S1)-axis, showing the rotation of asensitive axis u₁ of the sensor S1 about the z″_(S1)-axis;

FIG. 9D is an orthogonal view of the x″y″-plane of the X″_(S2)-frame ofFIG. 9A, looking along the −z″_(S2)-axis, showing the rotation of asensitive axis u₂ of the sensor S2 about the z″_(S2)-axis;

FIG. 10A is a perspective view of a sensor trajectory of a sensor movingwith respect to a first reference frame X of an instrument according tosome embodiments, a second reference frame X′ orbits an origin of thefirst reference frame X, and a third reference frame X″ orbits an originof the second reference frame X′ in a plane of rotation which istangential to the trajectory of the second reference frame X;

FIG. 10B is a plan view of the trajectory of FIG. 10A;

FIG. 10C is a side view of the trajectory of FIG. 10A;

FIG. 10D is an orthogonal view of the z′x′-plane of FIG. 10A showing therelative motion of the third reference frame X″ with respect to thesecond reference frame X′;

FIG. 11 is a schematic diagram of a measuring instrument according tosome embodiments;

FIG. 12 is a schematic diagram of a drilling system comprising ameasuring instrument according to some embodiments;

FIG. 13A is a plan view of a mechanism and components of a downholemeasuring system according to some embodiments;

FIG. 13B is a first cross-section of the mechanism of FIG. 13A; and

FIG. 13C is a second cross-section of the mechanism of FIG. 13A.

DESCRIPTION OF EMBODIMENTS

Embodiments generally relate to measuring instruments, systems andmethods, and in particular to magnetic gradiometers.

Magnetic field sensors are of two general types. Uniaxial or directionaltype sensors can measure a single magnetic field component along asensitive axis of the sensor. This type includes fluxgate magnetometers,superconducting quantum interference devices (SQUIDs), Hall effectmagnetometers and various magnetoresistive devices, for example. Anothertype of sensor, known as a scalar or total field magnetometer, measuresthe total magnitude of the magnetic field vector or field strength.

The gradient of a particular magnetic field component can be determinedby subtracting measurements from two magnetic field sensors that areseparated by a baseline. If the distance to the nearest magnetic sourceis substantially greater than the baseline between the sensors of thegradiometer, the difference in the measured magnetic field, divided bythe baseline, is approximately equal to the magnetic field gradient in adirection of the baseline. Arrays of magnetic field sensors in suitableconfigurations can be used to define gradients of multiple magneticfield components along multiple directions.

Some sensors can be designed as intrinsic gradiometers, with outputsthat are proportional to the difference in a magnetic field componentacross a small baseline distance. For example, axial and planarintrinsic gradiometers can be constructed from pairs of suitablyoriented and positioned adjacent superconducting loops, connected suchthat the superconducting currents induced within each loop, which areproportional to the magnetic flux through each loop, are opposed to eachother. In theory, this produces cancellation of the signal in a uniformfield, and a signal proportional to the gradient in a non-uniform field,which can be read using a SQUID sensor. Higher order gradients can bemeasured by more complex arrangements of superconducting pickup loops.

The nine elements of the magnetic gradient tensor G consist of gradientsof all three orthogonal vector field components along all threecorresponding spatial directions. For example the spatial derivativealong the x axis of the y component of the field vector is the gradienttensor element G_(xy). The gradient tensor G is traceless(G_(xx)+G_(yy)+G_(zz)=0). Additionally, in a non-conducting medium, suchas air, Maxwell's equations imply that the gradient tensor associatedwith a quasistatic magnetic field is symmetric (G_(yx)=G_(xy)). In thiscase, the gradient tensor has only five independent elements.Measurement of the full gradient tensor at a single location thereforeprovides five independent pieces of information about the structure ofthe magnetic field at that point, whereas the magnetic field vectorB=(B_(x), B_(y), B_(z)) provides three pieces of information and thetotal field T=(B_(x) ²+B_(y) ²+B_(z) ²)^(1/2) provides only a singlepiece of information. As a result measurements of the gradient tensor ata limited number of locations may provide improved localisation ofmagnetic sources and better characterisation of their magneticproperties, compared to measurements of only the magnetic field vectoror the total field.

Measurements of the magnetic field vector made on a moving platform maybe adversely affected by motion noise, as small changes in orientationof the platform as it moves through the geomagnetic field, which isgenerally much larger than the small field variations of interest,produce variations in the measured field components that are comparableto, or even much larger than, the signal from the magnetic sourcetargets of interest. Motion noise is much less significant for magneticgradient measurements, because the background gradient of thegeomagnetic field is relatively small compared to the gradients due tolocal target sources. This is one advantage of gradient measurementsover vector field measurements.

One difficulty in determining the magnetic gradient tensor using anarray of vector magnetic sensors is that small differences incalibration constants and slight misalignment of sensitive axes ofsensors imply that a gradiometer array in practice measures acombination of the field (the common-mode signal) and its gradient.Overcoming this problem requires careful calibration, which may have tobe repeated frequently, and independent measurement of the field by aseparate referencing magnetometer.

Subtracting the outputs of a pair of suitably calibrated paralleluniaxial magnetometers, mounted side-by-side on a rotating disc, canprovide the off-diagonal gradient tensor element G_(x′y′), where x′ isthe instantaneous sensitive axis and y′ lies along the baseline joiningthe centres of the two magnetometers. When such a gradiometer rotates ina field with a uniform gradient, the sum of the magnetometer outputswill vary sinusoidally with rotation angle, with the amplitude beingproportional to the field component B=(B _(x),B _(y)) in the plane ofthe disc, averaged over the disc. The same type of signal at thefundamental rotation frequency results from the residual subtractedoutput from imperfectly balanced sensors (i.e. the common mode signal).With respect to a fixed direction x in the rotation plane, the in-phasecomponent of the fundamental harmonic is proportional to the x componentof the field and the quadrature signal is proportional to the ycomponent. The second harmonic component of the subtracted outputscomprises an in-phase component proportional to G_(xy) and a quadraturecomponent proportional to (G_(xx)−G_(yy)).

One approach to overcoming this difficulty is to rotate a gradiometer inthe field to be measured, which can average out the common mode signal,in principle leaving a pure gradient signal. However, a serious problemwith a rotating dual sensor gradiometer is caused by interaction betweenthe sensors. For example for fluxgate sensors, interactions include bothinductive coupling at the excitation frequency and its harmonics, andmagnetostatic anomalies arising from each ferromagnetic core thatperturb the field at the other. As the gradiometer rotates theferromagnetic cores act as magnetic sources with highly anisotropic, andprobably nonlinear, permeability, responding to a rotating appliedfield.

Undesirable interactions may be eliminated altogether by removing one ofthe sensors and analysing the signal from a single sensor as it followsa circular trajectory, which may be about the centre of a non-conductingand non-magnetic disc on which it is mounted, for example. Fourieranalysis of the resulting signal can isolate signals due to the fieldcomponents and gradient components to allow determination of multipleelements of the magnetic gradient tensor.

Referring to FIG. 1, a measuring instrument 100 is represented as ablock diagram. The instrument 100 comprises: a sensor 110 mounted on amechanism 120 to move the sensor 110 relative to the instrument 100, asignal processor 130 to receive a measurement signal from the sensor110, and a power source 140 connected to the signal processor 130 andmechanism 120 to power the signal processor 110 and drive the motion ofthe sensor 110. The instrument 100 may further comprise: an actuator 150coupled to the mechanism 120 to drive the mechanism 120 to move thesensor 110, a controller 160 connected to the actuator 150 to controlthe actuator 150, a computer processor 170 connected to the controller160 and the signal processor 130 to monitor and control the controller160 and the signal processor 130, and one or more angular positionsensors 180 coupled to the mechanism 120 to measure the angular positionand/or orientation of the sensor 110. The power source 140 may also beconnected to the actuator 150, controller 160 and computer processor170.

The mechanism 120 is adapted to move the sensor 110 in a predeterminedpath or trajectory relative to a fixed reference frame of the instrument100 such that the sensor 110 measures a particular property of the localenvironment of the instrument 100 such as the magnetic field, forexample and the signal measured along the path can then be analysed tocompare the property at various points along the path to determinespatial gradients of the property in the local environment of theinstrument 100. The predetermined path of the sensor 110 may bemathematically defined and known to allow comparison of the sensorsignal with the predetermined path. If the predetermined path is knownby a user, the user can analyse the sensor signal and compare it againstthe path of the sensor 110 to determine spatial gradients of themeasured property. Alternatively, the predetermined path may be enteredinto the computer processor 170 to allow the computer processor 170 toanalyse the sensor signal and compare it against the path of the sensor110 to determine one or more characteristics of the measured property,such as spatial gradients, for example.

The predetermined path of the sensor 110 may comprise one or morerotational components having angular velocities and associated frequencycomponents. The sensor signal may also comprise frequency componentsrelated to the predetermined path of the sensor 110. The frequencycomponents of the sensor signal may be identified and compared with thefrequency components associated with the predetermined path of thesensor 110 to determine one or more characteristics of the property ofinterest.

The sensor 110 may be adapted to measure various properties of the localenvironment of the measuring instrument 100 such as physical propertiesincluding scalar quantities such as total magnetic intensity;directional quantities such as individual components of magnetic,electric or gravitational fields; or vector quantities such as magneticfields, electric fields, gravitational or other force fields, forexample. In some embodiments, the sensor 110 may comprise multiplesensors to measure different properties simultaneously, or multiplesensors to measure the same property to allow for comparison and/orcalibration.

In some embodiments, the sensor 110 may comprise a single magnetometeror magnetic field sensor and the instrument 100 may comprise a magneticgradiometer, which may also provide measurements of magnetic fieldvector components. Some magnetometers which may be used are:directionally sensitive magnetometers, uniaxial magnetometers, fluxgatemagnetometers, superconducting quantum interference devices (SQUIDs),Hall effect magnetometers, various magnetoresistive devices, anisotropicmagnetoresistance (AMR) magnetometers, scalar magnetometers or totalfield magnetometers such as chip-scale atomic magnetometers, forexample. The motion of the sensor 110 relative to the instrument 100 mayallow multiple components of the magnetic gradient tensor to be measuredin the local environment of the instrument 100, such as one, two, three,four or five independent components of the magnetic gradient tensor, forexample. As well as measuring magnetic gradients, the motion of thesensor 110 relative to the instrument 100 may allow multiple componentsof the magnetic field vector to be measured in the local environment ofthe instrument 100, such as one, two, or three, independent componentsof the magnetic field vector, for example. The motion of the sensor 110relative to the instrument 100 may allow multiple components of thevector gradient of the total magnetic intensity to be measured in thelocal environment of the instrument 100, such as one, two, or threeindependent components of the vector gradient of the total magneticintensity, for example.

Some exemplary applications for the magnetic gradiometers describedherein include: measurement-while-drilling (MWD) of magnetic fields andgradients in boreholes to detect external magnetic sources or to studymagnetisation of rock formations; detection, localisation andclassification (DLC) of magnetic targets from static or mobileplatforms; downhole magnetic surveys for mineral exploration orenvironmental studies; mapping of magnetic field anomalies forgeological interpretation, resource exploration and in environmental andarchaeological surveys; detection of unexploded ordnance (UXO); anddetecting or measuring other magnetic targets of interest, such as landmines, naval mines, submarines, shipwrecks, archaeological artefacts andstructures, buried drums containing toxic waste, for example, and manyothers. Apart from their uses in systematic magnetic surveys, gradienttensor measurements have a specific application to manoeuvrable searchplatforms that home onto compact magnetic targets, such as buried landmines, naval mines and UXO. Other applications include fixed magneticgradiometers that are used to track moving sources such as submarinesentering a harbour, concealed handguns carried into a room, hypodermicneedles within hospital laundry, or ferrous contaminants in foodstuffsmoving along a conveyor belt, for example.

In embodiments where the sensor 110 comprises a directionally sensitivesensor, an axis along which the sensor 110 is most sensitive may bereferred to as the sensitive axis and may be denoted by vector u.

The power source 140 may supply power to the signal processor 130, theactuator 150, the controller 160, and/or the computer 170. The powersource 140 may also supply power to the sensor 110 if necessary.

The signal processor 130 may receive and process a signal measured bythe sensor 110, which may be referred to as a sensor signal or amagnetometer signal in embodiments where the sensor 110 comprises amagnetometer. The signal processor 130 may then determine certain localenvironmental properties or physical properties based on measurementsfrom the sensor 110. Alternatively, the signal processor 130 may processthe measured signal and output a processed signal to be received andanalysed by the computer 170.

Frequency components of the signal measured by the sensor 110 may beanalysed by an analogue filter bank or by digital sampling. The signalmay be sampled by an analogue-to-digital converter (ADC) 135, at equalincrements of the rotation angle(s), as determined from a signalgenerated by the angular position sensor(s) 180. Alternatively, if theangular speed of the rotary motion is kept constant, the signal may besampled at equal time intervals. The angular position sensor 180 maycomprise any suitable angular position sensor such as a shaft encoder,or an optical sensor that detects fiducial marks or slots in a rotor,for example.

The digital samples of the signal measured by the sensor 110 may then beinput to a discrete Fourier transform, which may be implemented eitherby hardware, such as by a fast Fourier transform (FFT) chip 137 whichmay form part of the signal processor 130, or by standard FourierTransform software executed by the computer 170. The output of theFourier analysis may comprise amplitudes and phases (or in-phase andquadrature components) of the discrete frequency components of thesignal measured by the sensor 110, where the phase is determinedrelative to an output signal of the angular position sensor(s) 180.

The output of the Fourier analysis can then be compared against thetrajectory of the sensor 110 to determine a number of elements of thegradient tensor as well as the directional components of the fieldstrength if the measured field is a vector field, or a number of spatialgradients of the field intensity if the measured field is the scalarintensity. In the case of magnetic field measurement, elements of themagnetic gradient tensor may be determined where the magnetic gradienttensor is defined as:

${G = {G_{ij} = \frac{\partial B_{x_{i}}}{\partial x_{j}}}},$

where B is the magnetic field, x_(i) denotes directions (x, y, z) andB_(xi) denotes the magnetic field components (B_(x), B_(y), B_(z)).

The output of the Fourier analysis of a signal measured by adirectionally sensitive magnetometer 110 can be compared against thetrajectory of the sensor 110 to determine some or all of the elements ofthe magnetic gradient tensor G as well as some or all of the componentsof the magnetic field B.

Fourier analysis of the signal measured by a directionally sensitivemagnetometer 110 can be compared against the trajectory of the sensor110 to also determine some or all of the higher order gradients of themagnetic field B from frequency components that are higher multiples ofthe rotation frequency. For example, second order gradients of themeasured field, which are components of a third order tensor,grad(gradB), can be extracted from the third harmonic of the rotationfrequency.

The actuator 150 may comprise one or more actuators such as electricmotors, for example, disposed to apply one or more input forces to themechanism 120 to drive the mechanism 120 and move the sensor 110 in thepredetermined path.

The actuator 150 may be controlled by the controller 160 in order todrive the mechanism 120 in a predetermined manner to move the sensor 110in a required direction and at a required speed. In some embodiments,the controller 160 may control the actuator 150 by controlling powersupply from the power source 140 to the actuator 150.

The controller 160 may be controlled or operated by the computer 170executing program code accessible by the computer 170 to cause thecontroller 160 to issue control signals to the actuator 150.

The computer 170 may also control or operate the signal processor 130,or at least receive an output signal from the signal processor 130 andanalyse the output signal to determine one or more spatial gradients ofthe measured quantity. In embodiments where the instrument 110 comprisesa magnetic gradiometer, the computer 170 may determine one or moreoutput measurements selected from: the total magnetic intensity, anaverage of a component of the magnetic field strength in a directionalong an x-axis of the instrument 100, an average of a component of themagnetic field strength in a direction along a y-axis of the instrument100, an average of a component of the magnetic field strength in adirection along a z-axis of the instrument 100, the gradient of the xcomponent in the x direction, the gradient of the y component in the ydirection, the difference between the gradient of the x component in thex direction and the gradient of the y component in the y direction, thegradient of the z component in the z direction, the gradient of the xcomponent in the y direction, the gradient of the y component in the xdirection, the gradient of the y component in the z direction, thegradient of the z component in the y direction, the gradient of the xcomponent in the z direction, and the gradient of the z component in thex direction.

The instrument may further comprise a user interface 175 for a user tocontrol the computer 170 and receive output measurements determined bythe computer 170. The user interface 175 may comprise a display todisplay the one or more output measurements determined by the computer170.

In some embodiments, the computer 170 and/or signal processor 160 may beremote from the sensor 110 and mechanism 120. In some embodiments, themeasured signal may be recorded and stored on a memory (not shown) forlater analysis by a computer 170.

The mechanism 120 may cause the sensor 110 to move relative to the fixedreference frame of the instrument 100 in any suitable path that can beinput into the computer 170 to allow the computer 170 to determine theone or more output measurements based on the measured or processedsignal and the trajectory of the sensor 110.

Some exemplary sensor trajectories are shown in FIGS. 2A to 10Daccording to some embodiments. Equations are presented for the value ofa quantity B_(u) measured by a uniaxial sensor 110 with sensitive axis u(such as a uniaxial magnetic sensor, for example) moving through avector field B (such as a magnetic field, for example) along the varioustrajectories presented in each of FIGS. 2A to 10D. Each of thetrajectories described below include one or two rotation components withfirst and second associated angular velocities ω₁ and ω₂. In someembodiments, the angular velocity ω₁ may be constant or uniform. Inother embodiments, the angular velocity ω₁ may be variable with time. Insome embodiments, the angular velocity ω₂ may be constant or uniform. Inother embodiments, the angular velocity ω₂ may be variable with time.

If an angular velocity is constant or uniform, the sensor signal can besampled at equal time intervals and the frequency components of thesignal will correspond to the temporal frequency of the angularvelocity. The frequency components can then be identified throughFourier analysis, for example, where the Fourier components correspondto the temporal frequency components.

If an angular velocity is variable and not uniform, the sensor signalcan be sampled at equal angular increments of the rotations, rather thanat equal time intervals. In these cases, the Fourier componentsrepresent cycles per rotation, rather than angular frequencies in unitssuch as rad/s or frequencies in Hz. Throughout this document the term“frequency component” represents a component of a sensor signal, or acomponent associated with a predetermined path or trajectory, defined incycles per rotation. The frequency components of a sensor signal may beisolated by Fourier analysis of the sensor signal, which is readilyconverted into units of rad/s or Hz if the rotation rates are uniform.

FIGS. 2A to 2D show a circular sensor trajectory with the sensitive axisu directed tangentially to the trajectory. A first fixed reference frameX of the instrument 100 is shown as orthogonal coordinate axes X=(x,y,z)having an origin O at the centre. A second orbiting reference frameX′=(x′,y′,z′) rotates about the origin O with an origin O′ of thereference frame X′ following a circular trajectory described by r₁ at adistance |r₁|=a/2 from the origin O.

Reference frame X′ rotates such that: x′ is directed in the direction ofmotion of X′ tangential to the trajectory of X; y′ is directed radiallyinward towards O; and z′ remains parallel to and spaced from z. Theangular velocity of X′ orbiting O is denoted by ω₁, and the angle ofrotation θ is defined as the angle between x and x′ (θ=0 when x′ isparallel to x, θ is defined as positive as measured from x towards y,and θ=ω₁t, where t=time, for the case of uniform rotational motion). Insome embodiments, the angular velocity ω₁ may be constant. In otherembodiments, the angular velocity ω₁ may not be constant.

The sensor 110 is located at O′ and the direction of the sensitive axisu fixed relative to X′. In various embodiments, the sensitive axis u maybe directed in any desired direction with respect to X′; however, inFIGS. 2A to 2D, the sensitive axis u is aligned with x′.

r ₁=(x,y), where x=(a/2)sin θ, y=−(a/2)cos θ.  (1)

Assuming a uniform field gradient, the field vector in the plane ofrotation at this location is given by

$\begin{matrix}\begin{matrix}{{B( {x,y} )} = {{B( {0,0} )} + {G \cdot r}}} \\{= {\overset{\_}{B} + {( {a/2} )( {{G_{xx}\sin \; \theta} - {G_{xy}\cos \; \theta}} )\hat{x}} +}} \\{{( {a/2} )( {{G_{xy}\sin \; \theta} - {G_{yy}\cos \; \theta}} )\hat{y}}}\end{matrix} & (2)\end{matrix}$

where B=(B _(x), B _(y)) is the field component in the xy-plane averagedover the sensor trajectory. In equation (2), and throughout thisdocument, unit vectors are denoted by hats above bold symbols thatdenote vector quantities.

The component of this field B measured by the sensor is

$\begin{matrix}\begin{matrix}{B_{u} = {{B_{x}\cos \; \theta} + {B_{y}\sin \; \theta}}} \\{= {{{\overset{\_}{B}}_{x}\cos \; \theta} + {{\overset{\_}{B}}_{y}\sin \; \theta} - {( {a/2} ){G_{xy}( {{\cos^{2}\theta} - {\sin^{2}\theta}} )}} +}} \\{{( {a/2} )( {G_{xx} - G_{yy}} )\sin \; \theta \; \cos \; \theta}} \\{= {{{\overset{\_}{B}}_{x}\cos \; \theta} + {{\overset{\_}{B}}_{y}\sin \; \theta} - {( {a/2} )G_{xy}\cos \; 2\theta} +}} \\{{( {a/4} )( {G_{xx} - G_{yy}} )\sin \; 2{\theta.}}}\end{matrix} & (3)\end{matrix}$

Equation (3) shows that, as the sensor 110 orbits the origin O, thein-phase and quadrature signals at the fundamental rotation rate areproportional to the x and y field components at the origin Orespectively, and the in-phase and quadrature components of the secondharmonic signal are, respectively, proportional to G_(xy) and(G_(xx)−G_(yy)), which are mathematically the two independent componentsof the differential curvature of the magnetic scalar potential.

One advantage of a magnetic gradiometer with a single magnetic sensor110 is that only one calibration constant is required to convert thesignal measured by the sensor 110 (in volts, for example) into field andgradient components in nT and nT/m respectively (or any other suitableunits for magnetic strength). Measurements about three suitably orientedrotation axes should provide sufficient information to calculate thefull magnetic field vector and magnetic gradient tensor.

The single fixed orbiting uniaxial magnetometer may also provide anadvantage due to its simplicity, and may provide a clean separation offield and gradient components in the frequency domain. Separation offield and gradient signals into different frequency bins greatly reducescontamination of the gradient signal by misalignment of the sensitiveaxes of multiple sensors and by vibration.

On the other hand, a single sensor system only gives partial informationabout the gradient tensor, so complete determination of the magneticgradient tensor in a down-hole survey requires three such systems withrotation axes that are oblique to the drilling axis, or a singleobliquely oriented system for which the rotation axis precesses aboutthe drilling axis (or instrument z-axis).

In some embodiments, the uniaxial magnetometer may be supplemented bymagnetometers with sensitive axes aligned parallel to y′ (radiallyinward), and z′ (to form an orthogonal right-handed system). As themagnetometer orbits the instrument z axis, the signal from thetangential sensor is given by (3) and the signals from the other twosensors are given by:

$\begin{matrix}\begin{matrix}{B_{y^{\prime}} = {{{- B_{x}}\sin \; \theta} + {B_{y}\cos \; \theta}}} \\{= {{{- {\overset{\_}{B}}_{x}}\sin \; \theta} + {{\overset{\_}{B}}_{y}\cos \; \theta} - {( {a/2} )( {{G_{xx}\sin^{2}\theta} - {G_{yy}\cos^{2}} - {G_{xy}\sin \; \theta \; \cos \; \theta}} )}}} \\{= {\frac{{- ( {G_{xx} + G_{yy}} )}a}{4} - {{\overset{\_}{B}}_{x}\sin \; \theta} + {{\overset{\_}{B}}_{y}\cos \; \theta} + {\frac{( {G_{xx} - G_{yy}} )a}{4}\cos \; 2\theta} +}} \\{{( {G_{xy}{a/2}} )\sin \; 2\theta}} \\{= {\frac{G_{zz}a}{4} - {{\overset{\_}{B}}_{x}\sin \; \theta} + {{\overset{\_}{B}}_{y}\cos \; \theta} + {\frac{( {G_{xx} - G_{yy}} )a}{4}\cos \; 2\theta} + {( {G_{xy}{a/2}} )\sin \; 2{\theta.}}}}\end{matrix} & (8) \\{\mspace{20mu} {and}} & \; \\{{B_{z}( {x,y} )} = {{{B_{z}( {0,0} )} + \frac{\partial B_{z}}{\partial x} + {\frac{\partial B_{z}}{\partial y}y}} = {{\overset{\_}{B}}_{z} + {( {G_{xz}{a/2}} )\sin \; \theta} - {( {G_{yz}{a/2}} )\cos \; {\theta.}}}}} & (9)\end{matrix}$

Comparing (8) with (3), it is clear that the measured radial componentsupplies equivalent information to the tangential component,supplemented by a DC term that is proportional toG_(zz)=(G_(xx)+G_(yy)). In principle, knowledge of the sum anddifference of G_(xx) and G_(yy), from the DC and in-phase secondharmonic terms respectively, then allows determination of all of thediagonal components of the gradient tensor. However, any unknown offsetin the sensor will contaminate the DC term and introduce errors into thecalculated gradients. In practice, the choice between tangential orradial orientation of a single orbiting sensor will usually bedetermined by convenience of layout or manufacture.

From (9) the z component augments the other two components withinformation about B_(z), G_(xz) and G_(yz). The tangential and radialcomponents each define the quantity G_(xx)−G_(yy), but do not provideenough information to determine the diagonal components of the tensoruniquely.

If two orbiting triaxial (or biaxial x′z′ or y′z′) magnetometer systemsare mounted along the instrument z axis, separated by a baseline z₀,then the field vector and gradient tensor midway between these twosensors are given by:

$\begin{matrix}{\mspace{20mu} {{{{\overset{\_}{B}}_{x}(z)} = \frac{{{\overset{\_}{B}}_{x}( {z - {z_{0}/2}} )} + {{\overset{\_}{B}}_{x}( {z + {z_{0}/2}} )}}{2}},\mspace{20mu} {{{\overset{\_}{B}}_{y}(z)} = \frac{{{\overset{\_}{B}}_{y}( {z - {z_{0}/2}} )} + {{\overset{\_}{B}}_{y}( {z + {z_{0}/2}} )}}{2}},\mspace{20mu} {{{\overset{\_}{B}}_{x}(z)} = \frac{{{\overset{\_}{B}}_{z}( {z - {z_{0}/2}} )} + {{\overset{\_}{B}}_{z}( {z + {z_{0}/2}} )}}{2}},\mspace{20mu} {{G_{xy}(z)} = \frac{{G_{xy}( {z - {z_{0}/2}} )} + {G_{xy}( {z + {z_{0}/2}} )}}{2}},{{G_{xz}(z)} = {\frac{{G_{xz}( {z - {z_{0}/2}} )} + {G_{xz}( {z + {z_{0}/2}} )}}{2} \approx \frac{{{\overset{\_}{B}}_{x}( {z + {z_{0}/2}} )} - {{\overset{\_}{B}}_{x}( {z - {z_{0}/2}} )}}{z_{0}}}},{{G_{yz}(z)} = {\frac{{G_{yz}( {z - {z_{0}/2}} )} + {G_{yz}( {z + {z_{0}/2}} )}}{2} \approx \frac{{{\overset{\_}{B}}_{y}( {z + {z_{0}/2}} )} - {{\overset{\_}{B}}_{y}( {z - {z_{0}/2}} )}}{z_{0}}}},\mspace{20mu} {G_{zz} = {{- ( {G_{xx} + G_{yy}} )} = \frac{{{\overset{\_}{B}}_{z}( {z + {z_{0}/2}} )} - {{\overset{\_}{B}}_{z}( {z - {z_{0}/2}} )}}{z_{0}}}},}} & (10)\end{matrix}$

Since G_(xx)−G_(yy) is given by (3) or (8) and G_(xx)+G_(yy) is given by(10), the diagonal components G_(xx) and G_(yy) can be calculatedindividually, yielding the complete gradient tensor G.

Comparison of the frequency content of the signals given in (3), (8) and(10) shows that any misalignment of the sensor z axis with the axis oforbital motion will contaminate the estimates of G_(xz) and G_(yz) dueto contributions at the fundamental frequency from B_(x) and B_(y).Because the field is so large relative to the changes in fieldcomponents across the baseline, a small misalignment can lead to largeerrors in estimating these gradient components. It may therefore bepreferable to estimate these tensor components from the differences ofB_(x) and B_(y) measured along a finite baseline parallel to z, as inthe expressions on the far right-hand side of equation (10).

In some embodiments, the sensor 110 may comprise a total magneticintensity sensor such as a compact scalar magnetometer, for example. Thetotal magnetic intensity (TMI) T at the location of a sensor 110orbiting the instrument z-axis in a circular trajectory is:

$\begin{matrix}{{T( {x,y} )} = {{{T( {0,0} )} + {{\nabla T} \cdot r}} = {{\overset{\_}{T} + {\frac{\partial T}{\partial x}x} + {\frac{\partial T}{\partial y}y}} = {\overset{\_}{T} + {( {a/2} ){( {{\frac{\partial T}{\partial x}\sin \; \theta} - {\frac{\partial T}{\partial y}\cos \; \theta}} ).}}}}}} & (11)\end{matrix}$

A DC component of the signal gives the TMI at the centre of the orbit(origin O) and the amplitude and phase of the signal at the orbitalfrequency yield the orthogonal gradients of the TMI within the xy-plane.If a second TMI sensor is placed directly antipodal to the first TMIsensor, at (x,y)=(a/2)(−sin θ, cos θ), then the DC term in equation (11)is unchanged, but the in-phase and quadrature components at the rotationrate change sign. This means that the average signal from the twosensors is the TMI at the centre of the orbit and the subtracted signal(first sensor signal minus the second sensor signal) reflects only thegradients, with ∂T/∂x=(1/a)(Q₁−Q₂) and ∂T/∂y=(1/a)(I₂−I₁), where thesubscripts refer to the signals from the first and second sensors S1 andS2 respectively.

If two such orbiting TMI sensors are mounted along the instrument zaxis, separated by a baseline z₀, then the TMI, its vector gradient, anda pair of second order gradients at the midway point along the orbitalaxis are given by:

$\begin{matrix}{{{\overset{\_}{T}(z)} = \frac{{\overset{\_}{T}( {z - {z_{0}/2}} )} + {\overset{\_}{T}( {z + {z_{0}/2}} )}}{2}},{{\frac{\partial\overset{\_}{T}}{\partial x}(z)} = {\frac{1}{2}\lbrack {{\frac{\partial\overset{\_}{T}}{\partial x}( {z - {z_{0}/2}} )} + {\frac{\partial\overset{\_}{T}}{\partial x}( {z + {z_{0}/2}} )}} \rbrack}},{{\frac{\partial\overset{\_}{T}}{\partial y}(z)} = {\frac{1}{2}\lbrack {{\frac{\partial\overset{\_}{T}}{\partial y}( {z - {z_{0}/2}} )} + {\frac{\partial\overset{\_}{T}}{\partial y}( {z + {z_{0}/2}} )}} \rbrack}},{{\frac{\partial\overset{\_}{T}}{\partial z}(z)} = \frac{{\overset{\_}{T}( {z + {z_{0}/2}} )} + {\overset{\_}{T}( {z - {z_{0}/2}} )}}{z_{0}}},{{\frac{\partial^{2}\overset{\_}{T}}{{\partial x}{\partial z}}(z)} = {\frac{1}{z_{0}}\lbrack {{\frac{\partial\overset{\_}{T}}{\partial x}( {z + {z_{0}/2}} )} - {\frac{\partial\overset{\_}{T}}{\partial x}( {z - {z_{0}/2}} )}} \rbrack}},{{\frac{\partial^{2}\overset{\_}{T}}{{\partial y}{\partial z}}(z)} = {{\frac{1}{z_{0}}\lbrack {{\frac{\partial\overset{\_}{T}}{\partial y}( {z + {z_{0}/2}} )} - {\frac{\partial\overset{\_}{T}}{\partial y}( {z - {z_{0}/2}} )}} \rbrack}.}}} & (12)\end{matrix}$

The sensor trajectory shown in FIGS. 3A to 3D is the same as thetrajectory shown in FIGS. 2A to 2D, however the sensitive axis u isrotating with respect to X′. The sensitive axis u rotates in thex′z′-plane about y′ (and perpendicular to y′) with angular velocity ω₂and rotation angle φ (φ=0 when u is parallel to x′, φ is defined aspositive as measured between x′ and u from x′ towards z′, and φ=ω₂t,where t=time, for the case of uniform rotational motion). In someembodiments, the angular velocity ω₂ may be constant. In otherembodiments, the angular velocity ω₂ may not be constant.

The field component measured by the sensor 110 is given by:

B _(u) =B _(x′) cos φ+B _(z′) sin φ  (13)

Using (2) and (3), in terms of the field and gradient components withrespect to the instrument the measured field is therefore

$ {B_{u} = {{\lbrack {{{\overset{\_}{B}}_{x}\cos \; \theta} + {{\overset{\_}{B}}_{y}\; \sin \; \theta} - {( {a/2} )G_{xy}\cos \; 2\; \theta} + {( {a/4} )( {G_{xx} - G_{yy}} )\sin \; 2\; \theta}} \rbrack \cos \; \phi} + {{\overset{\_}{B}}_{z}\sin \; \phi} + {( {a/2} )G_{xz}\sin \; \theta \; \sin \; \phi} - {( {a/2} )G_{yz}\cos \; {\theta sin}\; \phi}}} \rbrack = {{\frac{{\overset{\_}{B}}_{x}}{2}\lbrack {{\cos \; ( {\theta - \phi} )} + {\cos ( {\theta + \phi} )}} \rbrack} + {\frac{{\overset{\_}{B}}_{y}}{2}\lbrack {{\sin ( {\theta - \phi} )} + {\sin ( {\theta + \phi} )}} \rbrack} + {{\overset{\_}{B}}_{z}\sin \; \phi} + {\frac{a}{2}\{ {{- {\frac{G}{2}\lbrack {{\cos ( {{2\theta} - \phi} )} + {\cos ( {{2\theta} + \phi} )}} \rbrack}} + {\frac{( {G_{xx} - G_{yy}} )}{4}\lbrack {{\sin ( {{2\theta} - \phi} )} + {\sin ( {{2\theta} + \phi} )}} \rbrack} + {\frac{G_{xz}}{2}\lbrack {{\cos ( {\theta - \phi} )} - {\cos ( {\theta + \phi} )}} \rbrack} + {\frac{G_{yz}}{2}\lbrack {{\sin ( {\theta - \phi} )} - {\sin ( {\theta + \phi} )}} \rbrack}} \}}}$

In terms of the rotation frequencies the measured field is therefore

$ {B_{u} = {{\lbrack {{{\overset{\_}{B}}_{x}\cos \; \omega_{1}t} + {{\overset{\_}{B}}_{y}\sin \; \omega_{1}t} - {( {a/2} )G_{xy}\cos \; 2\omega_{1}t} + {( {a/4} )( {G_{xx} - G_{yy}} )\sin \; 2\theta}} \rbrack \cos \; \phi} + {{\overset{\_}{B}}_{z}\sin \; \phi} + {( {a/2} )G_{xz}\sin \; {\theta sin}\; \phi} - {( {a/2} )G_{yz}\cos \; \theta \; \sin \; \phi}}} \rbrack = {{\frac{{\overset{\_}{B}}_{x}}{2}\lbrack {{{\cos ( {\omega_{1} - \omega_{2}} )}t} + {\cos ( {\omega_{1} + \omega_{2}} )}} \rbrack} + {\frac{{\overset{\_}{B}}_{y}}{2}\lbrack {{{\sin ( {\omega_{1} - \omega_{2}} )}t} + {\sin ( {\omega_{1} + \omega_{2}} )}} \rbrack} + {{\overset{\_}{B}}_{z}\sin \; \omega_{2}t} + {\frac{a}{2}\{ {{- {\frac{G_{xy}}{2}\lbrack {{{\cos ( {{2\omega_{1}} - \omega_{2}} )}t} + {{\cos ( {{2\omega_{1}} + \omega_{2}} )}t}} \rbrack}} + {\frac{( {G_{xx} - G_{yy}} )}{4}\lbrack {{{\sin ( {{2\omega_{1}} - \omega_{2}} )}t} + {{\sin ( {{2\omega_{1}} + \omega_{2}} )}t}} \rbrack} + {\frac{G_{xz}}{2}\lbrack {{{\cos ( {\omega_{1} - \omega_{2}} )}t} - {{\cos ( {\omega_{1} + \omega_{2}} )}t}} \rbrack} + {\frac{G_{yz}}{2}\lbrack {{{\sin ( {\omega_{1} - \omega_{2}} )}t} - {{\sin ( {\omega_{1} + \omega_{2}} )}t}} \rbrack}} \}}}$

Collecting terms and simplifying gives an expression for the Fouriercomponents in terms of field and gradient components

$\begin{matrix}{B_{u} = {{( {\frac{{\overset{\_}{B}}_{x}}{2} + \frac{G_{xz}a}{4}} )\lbrack {{{\cos ( {\omega_{1} - \omega_{2}} )}t} + {( {\frac{{\overset{\_}{B}}_{x}}{2} - \frac{G_{xz}a}{4}} ){\cos ( {\omega_{1} + \omega_{2}} )}t}} \rbrack} + {( {\frac{{\overset{\_}{B}}_{y}}{2} + \frac{G_{yz}a}{4}} )\lbrack {{{\sin ( {\omega_{1} - \omega_{2}} )}t} + {( {\frac{{\overset{\_}{B}}_{y}}{2} - \frac{G_{yz}a}{4}} ){\sin ( {\omega_{1} + \omega_{2}} )}t}} \rbrack} + {{\overset{\_}{B}}_{z}\sin \; \omega_{2}t} - {\frac{G_{xy}a}{4}{\cos ( {{2\omega_{1}} - \omega_{2}} )}t} + {\frac{G_{xy}a}{4}{\cos ( {{2\omega_{1}} + \omega_{2}} )}t} + {\frac{( {G_{xx} - G_{yy}} )a}{8}{\sin ( {{2\omega_{1}} - \omega_{2}} )}t} + {\frac{( {G_{xx} - G_{yy}} )a}{8}{\sin ( {{2\omega_{1}} + \omega_{2}} )}t}}} & (14)\end{matrix}$

It follows from (9) that all three components of the field vector, allthree off-diagonal components of the gradient tensor and the differencebetween the two diagonal tensor components in the plane of rotation aregiven by

$\begin{matrix}{{{\overset{\_}{B}}_{x} = {{I( {\omega_{1} - \omega_{2}} )} + {I( {\omega_{1} + \omega_{2}} )}}},{{\overset{\_}{B}}_{y} = {{Q( {\omega_{1} - \omega_{2}} )} + {Q( {\omega_{1} + \omega_{2}} )}}},{{\overset{\_}{B}}_{z} = {Q( \omega_{2} )}},{G_{xy} = \frac{2\lbrack {{I( {{2\omega_{1}} + \omega_{2}} )} - {I( {{2\omega_{1}} - \omega_{2}} )}} \rbrack}{a}},{G_{xz} = \frac{2\lbrack {{I( {\omega_{1} - \omega_{2}} )} - {I( {\omega_{1} + \omega_{2}} )}} \rbrack}{2}},{G_{yz} = \frac{2\lbrack {{Q( {\omega_{1} - \omega_{2}} )} - {Q( {\omega_{1} + \omega_{2}} )}} \rbrack}{a}},{{G_{xx} - G_{yy}} = \frac{4\lbrack {{Q( {{2\omega_{1}} - \omega_{2}} )} + {Q( {{2\omega_{1}} + \omega_{2}} )}} \rbrack}{2}},} & (15)\end{matrix}$

where I(ω), Q(ω) denote calibrated in-phase and quadrature Fouriercomponents at angular frequency ω.

If two of these rotating, orbiting magnetometer systems are mountedalong the instrument z axis, separated by a baseline z₀, then the fieldvector and gradient tensor midway between these two sensors are given by

$\begin{matrix}{\mspace{20mu} {{{{\overset{\_}{B}}_{x}(z)} = \frac{{{\overset{\_}{B}}_{x}( {z - {z_{0}/2}} )} + {{\overset{\_}{B}}_{x}( {z + {z_{0}/2}} )}}{2}},\mspace{20mu} {{{\overset{\_}{B}}_{y}(z)} = \frac{{{\overset{\_}{B}}_{y}( {z - {z_{0}/2}} )} + {{\overset{\_}{B}}_{y}( {z + {z_{0}/2}} )}}{2}},\mspace{20mu} {{{\overset{\_}{B}}_{x}(z)} = \frac{{{\overset{\_}{B}}_{z}( {z - {z_{0}/2}} )} + {{\overset{\_}{B}}_{z}( {z + {z_{0}/2}} )}}{2}},}} & (16) \\{\mspace{20mu} {{{G_{xy}(z)} = \frac{{G_{xy}( {z - {z_{0}/2}} )} + {G_{xy}( {z + {z_{0}/2}} )}}{2}},{{G_{xz}(z)} = {\frac{{G_{xz}( {z - {z_{0}/2}} )} + {G_{xz}( {z + {z_{0}/2}} )}}{2} = \frac{{{\overset{\_}{B}}_{x}( {z + {z_{0}/2}} )} - {{\overset{\_}{B}}_{x}( {z - {z_{0}/2}} )}}{2}}},{{G_{yz}(z)} = {\frac{{G_{yz}( {z - {z_{0}/2}} )} + {G_{yz}( {z + {z_{0}/2}} )}}{2} = \frac{{{\overset{\_}{B}}_{y}( {z + {z_{0}/2}} )} - {{\overset{\_}{B}}_{y}( {z - {z_{0}/2}} )}}{2}}},\mspace{20mu} {G_{zz} = {{- ( {G_{xx} + G_{yy}} )} = \frac{{{\overset{\_}{B}}_{z}( {z + {z_{0}/2}} )} - {{\overset{\_}{B}}_{z}( {z - {z_{0}/2}} )}}{z_{0}}}},}} & (17)\end{matrix}$

Since G_(xx)−G_(yy) and G_(xx)+G_(yy) are given by (15) and (17)respectively, the diagonal components G_(xx) and G_(yy) can becalculated individually, yielding the complete gradient tensor.Frequency domain separation yields reliable estimates of G_(xy) andG_(xx)−G_(yy), but estimates of G_(xz) and G_(yz) require accurateestimates of B_(x) and B_(y), which are adversely affected bymisalignment of sensor axes, because the signals of these gradientcomponents have the same frequency as the field components. However, thevector components derived from the sum signal, given by (16), provide areferencing signal which can be used to remove contamination of theG_(xz) and G_(yz) signals by imperfect alignment of the sensors. Thiscorrection can be determined by a calibration procedure, where a rangeof controlled uniform fields are applied to the instrument 100 and theapparent B_(xz) and B_(yz) gradient components measured. Thecoefficients of proportionality between the applied B_(x) and B_(y)fields and the apparent G_(xz) and G_(yz) gradients can be used tocorrect these measured gradient components for slight departures fromperfect alignment of the z′-axis with the instrument Z axis.

FIGS. 4A to 4D show a sensor arrangement and trajectories for twosensors S1 and S2 according to some embodiments. The two sensors S1 andS2 follow circular trajectories with the trajectory and sensor rotationof each sensor being similar to that described in relation to FIGS. 3Ato 3D. The sensors are located at opposite ends of an orbital diameter,orbiting the instrument z axis at angular velocity ω₁. Sensitive axes u₁and u₂ of respective sensors S1 and S2 rotate about an axis along thediameter of the orbit with angular velocity ω₂. In some embodiments, thesensors S1 and S2 may be inclined with respect to each other in theirrespective x′z′-planes and sensitive axes u₁ and u₂ may be offset by adifference in rotation angle Δφ in order to further reduce or mitigatemagnetic interference between the sensors. The difference in rotationangle Δφ may be any suitable angle between 0° and 180°, such as 90°, forexample. In other embodiments, the sensors S1 and S2 may be fixed inparallel relative to each other with Δφ=0°.

The measured fields in sensors S1 and S2, located at r₁ and −r₁respectively, are given by

B _(u) ₁ =B _(x) ₁ _(′) cos φ+B _(z) ₁ _(′) sin φ,

B _(u) ₂ =B _(x) ₂ _(′) cos φ+B _(z) ₂ _(′) sin φ  (18)

Using (2) and (3), in terms of the field and gradient components withrespect to the instrument the measured field is therefore

$\begin{matrix}{ {B_{u_{1}} = {{\lbrack {{{\overset{\_}{B}}_{x}\cos \; \theta} + {{\overset{\_}{B}}_{y}\sin \; \theta} - {( {a/2} )G_{xy}\cos \; 2\theta} + {( {a/4} )( {G_{xx} - G_{yy}} )\sin \; 2\theta}} \rbrack \cos \; \phi} + {{\overset{\_}{B}}_{z}\sin \; \phi} + {( {a/2} )G_{xz}\sin \; {\theta sin}\; \phi} - {( {a/2} )G_{yz}\cos \; \theta \; \sin \; \phi}}} \rbrack = {{\frac{{\overset{\_}{B}}_{x}}{2}\lbrack {{\cos ( {\theta - \phi} )} + {\cos ( {\theta + \phi} )}} \rbrack} + {\frac{{\overset{\_}{B}}_{y}}{2}\lbrack {{\sin ( {\theta - \phi} )} + {\sin ( {\theta + \phi} )}} \rbrack} + {{\overset{\_}{B}}_{z}\sin \; \phi} + {\frac{b}{2}\{ {{- {\frac{G_{xy}}{2}\lbrack {{\cos ( {{2\theta} - \phi} )} + {\cos ( {{2\theta} + \phi} )}} \rbrack}} + {\frac{( {G_{xx} - G_{yy}} )}{4}\lbrack {{\sin ( {{2\theta} - \phi} )} + {\sin ( {{2\theta} + \phi} )}} \rbrack} + {\frac{G_{xz}}{2}{ \quad{\lbrack {{\cos ( {\theta - \phi} )} - {\cos ( {\theta + \phi} )}} \rbrack + {\frac{G_{yz}}{2}\lbrack {{\sin ( {\theta - \phi} )} - {\sin ( {\theta + \phi} )}} \rbrack}} \}.}}} }}} & (19) \\{ {B_{u_{2}} = {{\lbrack {{{\overset{\_}{B}}_{x}\cos \; \theta} + {{\overset{\_}{B}}_{y}\sin \; \theta} + {( {a/2} )G_{xy}\cos \; 2\theta} - {( {a/4} )( {G_{xx} - G_{yy}} )\sin \; 2\theta}} \rbrack \cos \; \phi} + {{\overset{\_}{B}}_{z}\sin \; \phi} - {( {a/2} )G_{xz}\sin \; {\theta sin}\; \phi} + {( {a/2} )G_{yz}\cos \; {\theta sin}\; \phi}}} \rbrack = {{\frac{{\overset{\_}{B}}_{x}}{2}\lbrack {{\cos ( {\theta - \phi} )} + {\cos ( {\theta + \phi} )}} \rbrack} + {\frac{{\overset{\_}{B}}_{y}}{2}\lbrack {{\sin ( {\theta - \phi} )} + {\sin ( {\theta + \phi} )}} \rbrack} + {{\overset{\_}{B}}_{z}\sin \; \phi} + {\frac{b}{2}\{ {{\frac{G_{xy}}{2}\lbrack {{\cos ( {{2\theta} - \phi} )} - {\cos ( {{2\theta} + \phi} )}} \rbrack} - {\frac{( {G_{xx} - G_{yy}} )}{4}\lbrack {{\sin ( {{2\theta} - \phi} )} + {\sin ( {{2\theta} + \phi} )}} \rbrack} - {\frac{G_{xz}}{2}{ \quad{\lbrack {{\cos ( {\theta - \phi} )} - {\cos ( {\theta + \phi} )}} \rbrack - {\frac{G_{yz}}{2}\lbrack {{\sin ( {\theta - \phi} )} - {\sin ( {\theta + \phi} )}} \rbrack}} \}.}}} }}} & (20)\end{matrix}$

In terms of the rotation frequencies the sum and difference of themeasured outputs are given by:

$\begin{matrix}{{{B_{u_{1}} + B_{u_{2}}} = {{{{\overset{\_}{B}}_{x}\lbrack {{\cos ( {\theta - \phi} )} + {\cos ( {\theta + \phi} )}} \rbrack} + {{\overset{\_}{B}}_{y}\lbrack {{\sin ( {\theta - \phi} )} + {\sin ( {\theta + \phi} )}} \rbrack} + {2{\overset{\_}{B}}_{z}\sin \; \phi}} = {{{\overset{\_}{B}}_{x}\lbrack {{\cos ( {( {\omega_{1} - \omega_{2}} )t} )} + {\cos ( {( {\omega_{1} + \omega_{2}} )t} )}} \rbrack} + {{\overset{\_}{B}}_{y}\lbrack {{\sin ( {( {\omega_{1} - \omega_{2}} )t} )} + {\sin ( {( {\omega_{1} + \omega_{2}} )t} )}} \rbrack} + {2{\overset{\_}{B}}_{z}{\sin ( {\omega_{2}t} )}}}}},} & (21) \\{{B_{u_{1}} - B_{u_{2}}} = {{\frac{a}{2}\{ {{G_{xy}\lbrack {{\cos ( {\theta - \phi} )} - {\cos ( {{2\theta} + \phi} )}} \rbrack} - {\frac{( {G_{xx} - G_{yy}} )}{2}\lbrack {{\sin ( {{2\theta} - \phi} )} + {\sin ( {{2\theta} + \phi} )}} \rbrack} - {G_{xz}\lbrack {{\cos ( {\theta - \phi} )} - {\cos ( {\theta + \phi} )}} \rbrack} - {G_{yz}\lbrack {{\sin ( {\theta - \phi} )} - {\sin ( {\theta + \phi} )}} \rbrack}} \}} = {\frac{a}{2}\{ {{G_{xy}\lbrack {{\cos ( {( {{2\omega_{1}} - \omega_{2}} )t} )} - {\cos ( {( {{2\omega_{1}} + \omega_{2}} )t} )}} \rbrack} - {\frac{( {G_{xx} - G_{yy}} )}{2}\lbrack {{\sin ( {( {{2\omega_{1}} - \omega_{2}} )t} )} + {\sin ( {( {{2\omega_{1}} + \omega_{2}} )t} )}} \rbrack} - {G_{xz}\lbrack {{\cos ( {( {\omega_{1} - \omega_{2}} )t} )} - {\cos ( {( {\omega_{1} + \omega_{2}} )t} )}} \rbrack} - {G_{yz}\lbrack {{\sin ( {( {\omega_{1} - \omega_{2}} )t} )} - {\sin ( {( {\omega_{1} + \omega_{2}} )t} )}} \rbrack}} \}}}} & (22)\end{matrix}$

It follows from (21) that all three components of the field vector canbe extracted from the Fourier components of the sum of the signals:

B _(x) =I _(S)(ω₁−ω₂)=I _(S)(ω₁+ω₂)=[I _(S)(ω₁−ω₂)+I _(S)(ω₁+ω₂)]/2,

B _(y) =Q _(S)(ω₁−ω₂)=Q _(S)(ω₁+ω₂)=[Q _(S)(ω₁−ω₂)+Q _(S)(ω₁+ω₂)]/2,

B _(z) =Q _(S)(ω₂)/2,  (23)

where the subscript S denotes the sum. These components are offset-freeand, if the rotation frequencies are chosen to ensure that the sidebandsat ω₁±ω₂ are above the 1/f noise corner, exhibit negligible drift andlow noise.

Similarly it follows from (22) that in principle all three off-diagonalcomponents of the gradient tensor and the difference between the twodiagonal tensor components in the plane of rotation can be calculatedfrom the Fourier components of the difference signal (denoted bysubscript D):

$\begin{matrix}{{G_{xy} = {\frac{2{I_{D}( {{2\omega_{1}} + \omega_{2}} )}}{a} = {{- \frac{2{I_{D}( {{2\omega_{1}} - \omega_{2}} )}}{a}} = \frac{\lbrack {{I( {{2\omega_{1}} + \omega_{2}} )} - {I( {{2\omega_{1}} - \omega_{2}} )}} \rbrack}{a}}}},{G_{xz} = {\frac{2{I_{D}( {\omega_{1} + \omega_{2}} )}}{a} = {{- \frac{2{I_{D}( {\omega_{1} - \omega_{2}} )}}{a}} = \frac{\lbrack {{I_{D}( {\omega_{1} + \omega_{2}} )} - {I_{D}( {\omega_{1} - \omega_{2}} )}} \rbrack}{a}}}},{G_{yz} = {\frac{2{Q_{D}( {\omega_{1} + \omega_{2}} )}}{a} = {{- \frac{2Q_{D}( {\omega_{1} - \omega_{2}} )}{a}} = \frac{\lbrack {{Q_{D}( {\omega_{1} + \omega_{2}} )} - {Q_{D}( {\omega_{1} - \omega_{2}} )}} \rbrack}{a}}}},{{G_{xx} - G_{yy}} = {{- \frac{4{Q_{D}( {{2\omega_{1}} - \omega_{2}} )}}{a}} = {{- \frac{4{Q_{D}( {{2\omega_{1}} + \omega_{2}} )}}{a}} = {\frac{- {2\lbrack {{Q_{D}( {{2\omega_{1}} - \omega_{2}} )} + {Q_{D}( {{2\omega_{1}} + \omega_{2}} )}} \rbrack}}{a}.}}}}} & (24)\end{matrix}$

Comparison of (23) and (24) shows that any imbalance of the two sensors,which would give rise to a common mode signal when their outputs aredifferenced, produces Fourier components at ω₁+ω₂, which contaminate thesignal from B_(xz) and B_(yz). However, the vector components derivedfrom the sum signal, given by (23), provide a referencing signal whichcan be used to remove contamination of the B_(xz) and B_(yz) signals byimperfect balance of the sensors. This correction can be determined by acalibration procedure, where a range of controlled uniform fields areapplied to the instrument and the apparent B_(xz) and B_(yz) gradientcomponents measured. The coefficients of proportionality between theapplied B_(x) and B_(y) fields and the apparent B_(xz) and B_(yz)gradients can be used to correct these measured gradient components forthe common mode signal due to applied fields.

If two of these dual rotating, orbiting magnetometer systems are mountedalong the instrument z axis, separated by a baseline z₀, then the fieldvector and gradient tensor midway between these two sensors are given byequations (16) and (17), allowing all components of the gradient tensorto be determined without cross-contamination of frequency components.The baseline z₀ should be sufficient to ensure that magnetic andelectronic interference between the systems is negligible. Increasingthe length of the baseline along z improves the sensitivity of thegradient measurements along z, but z₀ should be small compared to thedistance to the nearest magnetic sources to ensure that that themeasurement accurately determines a first order gradient, rather than afinite difference.

The sensor trajectory shown in FIGS. 5A to 5D is the same as thetrajectory shown in FIGS. 2A to 2D, however the sensitive axis u isrotating with respect to X′. The sensitive axis u rotates in thez′y′-plane about x′ (and perpendicular to x′) with angular velocity ω₂and rotation angle φ (φ=0 when u is parallel to y′, φ is defined aspositive as measured between y′ and u from y′ towards z′, and φ=ω₂t,where t=time).

The measured field is given by

B _(u) =B _(y′) cos φ+B _(z′) sin φ  (25)

Using (8) and (9), in terms of the field and gradient components withrespect to the instrument, the measured field is therefore

$B_{u} = {{{B_{y^{\prime}}\cos \; \phi} + {B_{z^{\prime}}\sin \; \phi}} = {{{\lbrack {{{- {\overset{\_}{B}}_{x}}\sin \; \theta} + {{\overset{\_}{B}}_{y}\cos \; \theta} + {G_{zz}{a/4}} + {( {a/4} )( {G_{xx} - G_{yy}} )\cos \; 2\theta} + {( {a/2} )G_{xy}\sin \; 2\theta}} \rbrack \cos \; \phi} + {\lbrack {{\overset{\_}{B}}_{z} + {( {a/2} )G_{xz}\sin \; \theta} - {( {a/2} )G_{yz}\cos \; \theta}} \rbrack \sin \; \phi}} = {{- {\frac{{\overset{\_}{B}}_{x}}{2}\lbrack {{\sin ( {\theta - \phi} )} + {\sin ( {\theta + \phi} )}} \rbrack}} + {\frac{{\overset{\_}{B}}_{y}}{2}\lbrack {{\cos ( {\theta - \phi} )} + {\cos ( {\theta + \phi} )}} \rbrack} + {{\overset{\_}{B}}_{z}\sin \; \phi} + {\frac{a}{4}{\{ {{G_{zz}\cos \; \phi} + {G_{xy}\lbrack {{\sin ( {{2\theta} - \phi} )} + {\sin ( {{2\theta} + \phi} )}} \rbrack} + {\frac{( {G_{xx} - G_{yy}} )}{2}\lbrack {{\cos ( {{2\theta} - \phi} )} + {\cos ( {{2\theta} + \phi} )}} \rbrack} + {G_{xz}\lbrack {{\cos ( {\theta - \phi} )} - {\cos ( {\theta + \phi} )}} \rbrack} + {G_{yz}\lbrack {{\sin ( {\theta - \phi} )} - {\sin ( {\theta + \phi} )}} \rbrack}} \}.}}}}}$

In terms of the rotation frequencies, the measured field is therefore

$B_{u} = {{- {\frac{{\overset{\_}{B}}_{x}}{2}\lbrack {{\sin ( {( {\omega_{1} - \omega_{2}} )t} )} + {\sin ( {( {\omega_{1} + \omega_{2}} )t} )}} \rbrack}} + {\frac{{\overset{\_}{B}}_{y}}{2}\lbrack {{\cos ( {( {\omega_{1} - \omega_{2}} )t} )} + {\cos( {( {\omega_{1} + \omega_{2}} )t} \rbrack} + {{\overset{\_}{B}}_{z}{\sin ( {\omega_{2}t} )}} + {\frac{a}{2}{\{ {{G_{zz}{\cos ( {\omega_{2}t} )}} + {\frac{G_{xy}}{2}\lbrack {{\sin ( {( {{2\omega_{1}} - \omega_{2}} )t} )} + {\sin ( {( {{2\omega_{1}} + \omega_{2}} )t} )}} \rbrack} + {\frac{( {G_{xx} - G_{yy}} )}{2}\lbrack {{\cos ( {( {{2\omega_{1}} - \omega_{2}} )t} )} + {\cos ( {( {{2\omega_{1}} + \omega_{2}} )t} )}} \rbrack} + {\frac{G_{xz}}{2}\lbrack {{\cos ( {( {\omega_{1} - \omega_{2}} )t} )} - {\cos ( {( {\omega_{1} + \omega_{2}} )t} )}} \rbrack} + {\frac{G_{yz}}{2}\lbrack {{\sin ( {( {\omega_{1} - \omega_{2}} )t} )} - {\sin ( {( {\omega_{1} + \omega_{2}} )t} )}} \rbrack}} \}.}}} }}$

Collecting terms and simplifying gives an expression for the Fouriercomponents in terms of field and gradient components

$\begin{matrix}{B_{u} = {{( {\frac{{\overset{\_}{B}}_{y}}{2} + \frac{G_{xz}a}{4}} ){\cos ( {( {\omega_{1} - \omega_{2}} )t} )}} + {( {\frac{{\overset{\_}{B}}_{y}}{2} - \frac{G_{xz}a}{4}} ){\cos ( {( {\omega_{1} + \omega_{2}} )t} )}} - {( {\frac{{\overset{\_}{B}}_{x}}{2} - \frac{G_{yz}a}{4}} ){\sin ( {( {\omega_{1} - \omega_{2}} )t} )}} - {( {\frac{{\overset{\_}{B}}_{x}}{2} + \frac{G_{yz}a}{4}} ){\sin ( {( {\omega_{1} + \omega_{2}} )t} )}} + {( {G_{zz}{a/4}} ){\cos ( {\omega_{2}t} )}} + {{\overset{\_}{B}}_{z}{\sin ( {\omega_{2}t} )}} + {\frac{( {G_{xx} - G_{yy}} )a}{8}{\cos ( {( {{2\omega_{1}} - \omega_{2}} )t} )}} + {\frac{( {G_{xx} - G_{yy}} )a}{8}{\cos ( {( {{2\omega_{1}} + \omega_{2}} )t} )}} + {\frac{G_{xy}a}{4}{\sin ( {( {{2\omega_{1}} - \omega_{2}} )t} )}} + {\frac{G_{xy}a}{4}{\sin ( {( {{2\omega_{1}} + \omega_{2}} )t} )}}}} & (26)\end{matrix}$

It follows from (26) that all three components of the field vector andall components of the gradient tensor are given by

$\begin{matrix}{{{\overset{\_}{B}}_{x} = {{- {Q( {\omega_{1} - \omega_{2}} )}} - {Q( {\omega_{1} + \omega_{2}} )}}},{{\overset{\_}{B}}_{y} = {{I( {\omega_{1} - \omega_{2}} )} + {I( {\omega_{1} + \omega_{2}} )}}},{{\overset{\_}{B}}_{z} = {Q( \omega_{2} )}},{G_{xx} = \frac{2\lbrack {{I( {{2\omega_{1}} - \omega_{2}} )} + {I( {\omega_{1} + \omega_{2}} )} - {I( \omega_{2} )}} \rbrack}{a}},{G_{xy} = \frac{2\lbrack {{Q( {{2\omega_{1}} - \omega_{2}} )} + {Q( {{2\omega_{1}} + \omega_{2}} )}} \rbrack}{a}},{G_{xz} = \frac{2\lbrack {{I( {\omega_{1} - \omega_{2}} )} - {I( {\omega_{1} + \omega_{2}} )}} \rbrack}{a}},{G_{yy} = \frac{- {2\lbrack {{I( \omega_{2} )} + {I( {{2\omega_{1}} - \omega_{2}} )} + {I( {{2\omega_{1}} + \omega_{2}} )}} \rbrack}}{a}},{G_{yz} = \frac{2\lbrack {{Q( {\omega_{1} - \omega_{2}} )} - {Q( {\omega_{1} + \omega_{2}} )}} \rbrack}{a}},{G_{zz} = {4{{I( \omega_{2} )}/{a.}}}}} & (27)\end{matrix}$

where I(ω), Q(ω) denote calibrated in-phase and quadrature Fouriercomponents at angular frequency ω. Frequency domain separation yieldsreliable estimates of G_(xx), G_(xy), G_(yy) and G_(zz), but estimatesof G_(xz) and G_(yz) may be contaminated by contributions from B_(y) andB_(y) at the same frequencies. This contamination can be mitigated bycalibration. If the instrument is subjected to a range of uniformapplied fields along x and y, correction coefficients that enableremoval of the apparent gradient components due to applied fieldcomponents can be determined. Alternatively, if the instrument isremoved to a place where there is an essentially uniform backgroundfield, such as the geomagnetic field in an area of essentiallynon-magnetic rocks, or at high altitude, calibration of the instrumentcan be achieved by measuring the apparent field and gradient signalswhile the instrument is in several different orientations, so that allthree background geomagnetic field components vary with respect to theinstrument reference frame, and solving, in a least-squares sense, anover-determined matrix equation for the correction coefficients. Thiscalibration method also pertains to the other embodiments described.

If two of these rotating, orbiting magnetometer systems are mountedalong the instrument z axis, separated by a baseline z₀, then the fieldvector and gradient tensor midway between these two sensors are given by(16) and (17). The remarks above regarding calibration and alignment fordual sensors, separated along the instrument z axis, that rotate aboutthe radius vector also apply to the case of rotation about a tangentialaxis.

FIGS. 6A to 6D show a sensor arrangement and trajectories for twosensors S1 and S2 according to some embodiments. The two sensors S1 andS2 follow circular trajectories with the trajectory and sensor rotationof each sensor being similar to that described in relation to FIGS. 5Ato 5D. The sensors are located at opposite ends of an orbital diameter,orbiting the instrument z axis at angular velocity ω₁. Sensitive axes u₁and u₂ of respective sensors S1 and S2 rotate about an axis x′ which istangential to the motion. In some embodiments, the sensors S1 and S2 maybe inclined with respect to each other in their respective y′z′-planesand sensitive axes u₁ and u₂ may be offset by a difference in rotationangle Δφ in order to further reduce or mitigate magnetic interferencebetween the sensors. The difference in rotation angle Δφ may be anysuitable angle between 0° and 180°, such as 90°, for example. In otherembodiments, the sensors S1 and S2 may be fixed in parallel relative toeach other with Δφ=0°.

The measured fields in sensors 1 and 2, located at r and −rrespectively, are given by

B _(u) ₁ =B _(y) ₁ _(′) cos φ+B _(z) ₁ _(′) sin φ,

B _(u) ₂ =B _(y) ₂ _(′) cos φ+B _(z) ₂ _(′) sin φ  (28)

Using (2), (8) and (9), in terms of the field and gradient componentswith respect to the instrument the measured fields are therefore

B _(u) ₁ =[−B _(x) sin θ+ B _(y) cos θ+G _(zz) a/4+(a/4)(G _(xx) −G_(yy))cos 2θ+(a/2)G _(xy) sin 2θ] cos φ+[ B _(z)+(a/2)G _(xz) sinθ−(a/2)G _(yz) cos θ] sin φ,

B _(u) ₂ =[−B _(x) sin θ+ B _(y) cos θ−G _(zz) a/4−(a/4)(G _(xx) −G_(yy))cos 2θ−(a/2)G _(xy) sin 2θ] cos φ+[ B _(z)−(a/2)G _(xz) sinθ+(a/2)G _(yz) cos θ] sin φ   (29)

Expressing products of trigonometric functions in terms of sums anddifferences gives

$\begin{matrix}{B_{u_{1}} = {{- {\frac{{\overset{\_}{B}}_{x}}{2}\lbrack {{\sin ( {\theta - \phi} )} + {\sin ( {\theta + \phi} )}} \rbrack}} + {\frac{{\overset{\_}{B}}_{y}}{2}\lbrack {{\cos ( {\theta - \phi} )} + {\cos ( {\theta + \phi} )}} \rbrack} + {{\overset{\_}{B}}_{z}\sin \; \phi} + {\frac{a}{4}\{ {{G_{zz}\cos \; \phi} + {G_{xy}\lbrack {{\sin ( {{2\theta} - \phi} )} + {\sin ( {{2\theta} + \phi} )}} \rbrack} + {\frac{( {G_{xx} - G_{yy}} )}{2}\lbrack {{\cos ( {{2\theta} - \phi} )} + {\cos ( {{2\theta} + \phi} )}} \rbrack} + {G_{xz}{ \quad{\lbrack {{\cos ( {\theta - \phi} )} - {\cos ( {\theta + \phi} )}} \rbrack + {G_{yz}\lbrack {{\sin ( {\theta - \phi} )} - {\sin ( {\theta + \phi} )}} \rbrack}} \}.}}} }}} & (30) \\{B_{u_{2}} = {{- {\frac{{\overset{\_}{B}}_{x}}{2}\lbrack {{\sin ( {\theta - \phi} )} + {\sin ( {\theta + \phi} )}} \rbrack}} + {\frac{{\overset{\_}{B}}_{y}}{2}\lbrack {{\cos ( {\theta - \phi} )} + {\cos ( {\theta + \phi} )}} \rbrack} + {{\overset{\_}{B}}_{z}\sin \; \phi} - {\frac{a}{4}\{ {{G_{zz}\cos \; \phi} + {G_{xy}\lbrack {{\sin ( {{2\theta} - \phi} )} + {\sin ( {{2\theta} + \phi} )}} \rbrack} + {\frac{( {G_{xx} - G_{yy}} )}{2}\lbrack {{\cos ( {{2\theta} - \phi} )} + {\cos ( {{2\theta} + \phi} )}} \rbrack} + {G_{xz}{ \quad{\lbrack {{\cos ( {\theta - \phi} )} - {\cos ( {\theta + \phi} )}} \rbrack + {G_{yz}\lbrack {{\sin ( {\theta - \phi} )} - {\sin ( {\theta + \phi} )}} \rbrack}} \}.}}} }}} & (31)\end{matrix}$

In terms of the rotation frequencies the sum and difference of themeasured outputs are given by:

$\begin{matrix}{{B_{S} = {{B_{u_{1}} + B_{u_{2}}} = {{{{\overset{\_}{B}}_{y}\lbrack {{\cos ( {\theta - \phi} )} + {\cos ( {\theta + \phi} )}} \rbrack} - {{\overset{\_}{B}}_{x}\lbrack {{\sin ( {\theta - \phi} )} + {\sin ( {\theta + \phi} )}} \rbrack} + {2{\overset{\_}{B}}_{z}\sin \; \phi}} = {{{\overset{\_}{B}}_{y}\lbrack {{\cos ( {( {\omega_{1} - \omega_{2}} )t} )} + {\cos ( {( {\omega_{1} + \omega_{2}} )t} )}} \rbrack} - {{\overset{\_}{B}}_{x}\lbrack {{\sin ( {( {\omega_{1} - \omega_{2}} )t} )} + {\sin ( {( {\omega_{1} + \omega_{2}} )t} )}} \rbrack} + {2{\overset{\_}{B}}_{z}{\sin ( {\omega_{2}t} )}}}}}},} & (32) \\{B_{D} = {{B_{u_{1}} - B_{u_{2\;}}} = {\frac{a}{2}\{ {{{{( {G_{xx} - G_{yy}} )\lbrack {{\cos ( {{2\theta} - \phi} )} + {\cos ( {{2\theta} + \phi} )}} \rbrack}/2} + {G_{xy} \quad{\lbrack {{\sin ( {{2\theta} - \phi} )} + {\sin ( {{2\theta} + \phi} )}} \rbrack + {G_{xz}\lbrack {{\cos ( {\theta - \phi} )} + {\cos ( {\theta + \phi} )}} \rbrack} + {G_{yz}\lbrack {{\sin ( {\theta - \phi} )} - {\sin ( {\theta + \phi} )}} \rbrack}} \}}} = {\frac{a}{2}\{ {( {G_{xx} - G_{yy}} ){ \quad{{\lbrack {{\cos ( {( {{2\omega_{1}} - \omega_{2}} )t} )} + {\cos ( {( {{2\omega_{1}} + \omega_{2}} )t} )}} \rbrack/2} + {G_{xy}\lbrack {{\sin ( {( {{2\omega_{1}} - \omega_{2}} )t} )} + {\sin ( {( {{2\omega_{1}} + \omega_{2}} )t} )}} \rbrack} + {G_{xz}\lbrack {{\cos ( {( {\omega_{1} - \omega_{2}} )t} )} + {\cos ( {( {\omega_{1} + \omega_{2}} )t} )}} \rbrack} + {G_{yz}\lbrack {{\sin ( {( {\omega_{1} - \omega_{2}} )t} )} - {\sin ( {( {\omega_{1} + \omega_{2}} )t} )}} \rbrack}} \}.}} }} }}} & (33)\end{matrix}$

It follows from (32) that all three components of the field vector canbe extracted from the Fourier components of the sum of the signals:

B _(x) =−Q _(S)(ω₁−ω₂)=−Q _(S)(ω₁+ω₂)=−[Q _(S)(ω₁−ω₂)+Q _(S)(ω₁+ω₂)]/2,

B _(y) =I _(S)(ω₁−ω₂)=I _(S)(ω₁+ω₂)=[I _(S)(ω₁−ω₂)+I _(S)(ω₁+ω₂)]/2,

B _(z) =Q _(S)(ω₂)/2,  (34)

where the subscript S denotes the sum. These components are offset-freeand, if the rotation frequencies are chosen to ensure that the sidebandsat ω₁±ω₂ are above the 1/f noise corner, exhibit negligible drift andlow noise.

Similarly it follows from (33) that in principle all components of thegradient tensor can be calculated from the Fourier components of thedifference signal (denoted by subscript D):

$\begin{matrix}{{G_{xy} = {\frac{2{Q_{D}( {{2\omega_{1}} + \omega_{2}} )}}{a} = {\frac{2{Q_{D}( {{2\omega_{1}} - \omega_{2}} )}}{a} = \frac{\lbrack {{Q( {{2\omega_{1}} + \omega_{2}} )} + {Q( {{2\omega_{1}} - \omega_{2}} )}} \rbrack}{a}}}},{G_{xz} = {\frac{2{I_{D}( {\omega_{1} + \omega_{2}} )}}{a} = {\frac{2{I_{D}( {\omega_{1} - \omega_{2}} )}}{a} = \frac{\lbrack {{I_{D}( {\omega_{1} + \omega_{2}} )} + {I_{D}( {\omega_{1} - \omega_{2}} )}} \rbrack}{a}}}},{G_{yz} = {{- \frac{2{Q_{D}( {\omega_{1} + \omega_{2}} )}}{a}} = {\frac{2{Q_{D}( {\omega_{1} - \omega_{2}} )}}{a} = \frac{\lbrack {{Q_{D}( {\omega_{1} - \omega_{2}} )} - {Q_{D}( {\omega_{1} + \omega_{2}} )}} \rbrack}{a}}}},{{G_{xx} - G_{yy}} = {\frac{4{I_{D}( {{2\omega_{1}} - \omega_{2}} )}}{a} = {\frac{4{I_{D}( {{2\omega_{1}} + \omega_{2}} )}}{a} = \frac{2\lbrack {{I_{D}( {{2\omega_{1}} - \omega_{2}} )} + {I_{D}( {{2\omega_{1}} + \omega_{2}} )}} \rbrack}{a}}}},{G_{xx} = {\frac{( {G_{xx} - G_{yy}} ) - G_{zz}}{2} = \frac{\lbrack {{I_{D}( {{2\omega_{1}} - \omega_{2}} )} + {I_{D}( {{2\omega_{1}} + \omega_{2}} )} - {I_{D}( \omega_{2} )}} \rbrack}{a}}},{G_{yy} = {\frac{{- ( {G_{xx} - G_{yy}} )} - G_{zz}}{2} = \frac{- \lbrack {{I_{D}( {{2\omega_{1}} - \omega_{2}} )} + {I_{D}( {{2\omega_{1}} + \omega_{2}} )} + {I_{D}( \omega_{2} )}} \rbrack}{a}}},\mspace{20mu} {G_{zz} = {{- ( {G_{xx} + G_{yy}} )} = {\frac{2{I_{D}( \omega_{2} )}}{a}.}}}} & (35)\end{matrix}$

Comparison of (34) and (35) shows that any imbalance of the two sensors,which would give rise to a common mode signal when their outputs aredifferenced, produces Fourier components at ω₁±ω₂, which contaminate thesignal from B and B_(yz). However, the vector components derived fromthe sum signal, given by (34), provide a referencing signal which can beused to remove contamination of the B_(xz) and B_(yz) signals byimperfect balance of the sensors. This correction can be determined by acalibration procedure, where a range of controlled uniform fields areapplied to the instrument and the apparent B_(xz) and B_(yz) gradientcomponents measured. The coefficients of proportionality between theapplied B_(x) and B_(y) fields and the apparent B_(xz) and B_(yz)gradients can be used to correct these measured gradient components forthe common mode signal due to applied fields.

If two of these dual rotating, orbiting magnetometer systems are mountedalong the instrument z axis, separated by a baseline z₀, then the fieldvector and gradient tensor midway between these two sensors are given byequations (16) and (17), allowing all components of the gradient tensorto be determined without cross-contamination of frequency components.The alternative, second, expressions for G, and G_(yz) on the right handside of (17) and the expression for G_(zz) are unaffected by overlappingfrequencies, but require accurate alignment of xyz axes at each locationalong the z axis. To a first order approximation, in small azimuthalmisalignment angles, δ radians, between the xy axes at the two zlocations, the error in the estimated G_(xz) is proportional to B_(y)δand in the estimated G_(yz) is proportional to B_(x)δ. A similar remarkapplies to misalignment of z axes. These errors can be corrected bycalibration, applying a range of uniform fields along orthogonaldirections to determine the coefficients required to remove the effectsof misalignment, or by deploying the instrument in a region ofnegligible geomagnetic gradient and presenting it to the geomagneticfield in various orientations, as explained above.

The sensor trajectory shown in FIGS. 7A to 7E is the same as thetrajectory shown in FIGS. 2A to 2D, but the sensitive axis u is rotatingabout an inclined axis relative to X′. A third reference frameX″=(x″,y″,z″) is inclined and fixed with respect to X′. The inclinedframe X″ is rotated about y′ by an angle ψ such that y″=y′, there is anangle ψ between z″ and z′ and there is an angle ψ between x″ and x′. Theangle ψ is defined as positive as measured between x′ and x″ from x′towards z′.

The sensitive axis u is located at the X″ origin O″=O′=r₁ and rotates inthe x″y″-plane about z″ (and perpendicular to z″) with angular velocityω₂ and rotation angle φ (φ=0 when u is parallel to x″, φ is defined aspositive as measured between x″ and u from x″ towards y″, and φ=ω₂t,where t=time).

The field component measured by the sensor 110 is given by:

B _(u) =B _(x″) cos φ+B _(y″) sin φ,  (36)

where

B _(x″) =B _(x′) cos ψ+B _(z′) sin ψ=B _(x′) cos ψ+B _(z) sin ψ,  (37)

B _(x′) =B _(x) cos θ+B _(y) sin θ,  (38)

B _(y″) =B _(y′) =−B _(x) sin θ+B _(y) cos θ  (39)

Substituting (37)-(39) into (36) gives

$\begin{matrix}{B_{u} = {{{B_{x}\lbrack {{\cos \; \psi \; \cos \; \theta \; \cos \; \phi} - {\sin \; {\theta sin}\; \phi}} \rbrack} + {B_{y}\lbrack {{\cos \; \psi \; \sin \; {\theta cos}\; \phi} + {\cos \; {\theta sin\phi}}} \rbrack} + {B_{z}\sin \; {\psi cos}\; \phi}} = {{B_{x}\lbrack {{\frac{\cos \; \psi}{2}\lbrack {{\cos ( {\theta - \phi} )} + {\cos ( {\theta + \phi} )}} \rbrack} - {\frac{1}{2}\lbrack {{\cos ( {\theta - \phi} )} - {\cos ( {\theta + \phi} )}} \rbrack}} \rbrack} + {B_{y}\lbrack {{\frac{\cos \; \psi}{2}\lbrack {{\sin ( {\theta - \phi} )} + {\sin ( {\theta + \phi} )}} \rbrack} - {\frac{1}{2}\lbrack {{\sin ( {\theta - \phi} )} - {\sin ( {\theta + \phi} )}} \rbrack}} \rbrack} + {B_{z}\sin \; \psi \; \cos \; \phi}}}} & (40)\end{matrix}$

The field components at an instantaneous measurement point are

$\begin{matrix}{{{B_{x}( {x,y} )} = {{B_{x}( {{\frac{a}{2}\sin \; \theta},{{- \frac{a}{2}}\cos \; \theta}} )} = {{\overset{\_}{B}}_{x} + {( {a/2} )G_{xx}\sin \; \theta} - {( {a/2} )G_{xy}\cos \; \theta}}}},{{B_{y}( {x,y} )} = {{B_{y}( {{\frac{a}{2}\sin \; \theta},{{- \frac{a}{2}}\cos \; \theta}} )} = {{\overset{\_}{B}}_{y} + {( {a/2} )G_{xy}\sin \; \theta} - {( {a/2} )G_{yy}\cos \; \theta}}}},{{B_{z}( {x,y} )} = {{B_{z}( {{\frac{a}{2}\sin \; \theta},{{- \frac{a}{2}}\cos \; \theta}} )} = {{\overset{\_}{B}}_{z} + {( {a/2} )G_{xz}\sin \; \theta} - {( {a/2} )G_{yz}\cos \; \theta}}}}} & (41)\end{matrix}$

Substituting (40) into (41) and rearranging gives

$\begin{matrix}{B_{u} = {{\frac{{\overset{\_}{B}}_{x}}{2}\lbrack {{( {{\cos \; \psi} - 1} ){\cos ( {\theta - \phi} )}} + {( {{\cos \; \psi} + 1} ){\cos ( {\theta + \phi} )}}} \rbrack} + {\frac{{\overset{\_}{B}}_{y}}{2}\lbrack {{( {{\cos \; \psi} - 1} ){\cos ( {\theta - \phi} )}} + {( {{\cos \; \psi} + 1} ){\cos ( {\theta + \phi} )}}} \rbrack} + {{\overset{\_}{B}}_{z}\sin \; {\psi cos}\; \phi} + {\frac{a}{4}\lbrack {{G_{xx}\lbrack {{( {{\cos \; \psi} - 1} ){\cos ( {\theta - \phi} )}\sin \; \theta} + {( {{\cos \; \psi} + 1} ){\cos ( {\theta + \phi} )}\sin \; \theta}} \rbrack} - {G_{xy}\lbrack {{( {{\cos \; \psi} - 1} ){\cos ( {\theta - \phi} )}\cos \; \theta} + {( {{\cos \; \psi} + 1} ){\cos ( {\theta + \phi} )}\cos \; \theta} + {G_{xy}\lbrack {{( {{\cos \; \psi} - 1} )\sin \; ( {\theta - \phi} )\sin \; \theta} + {( {{\cos \; \psi} + 1} ){\sin ( {\theta + \phi} )}\sin \; \theta} - {G_{yy}\lbrack {{( {{\cos \; \psi} - 1} ){\sin ( {\theta - \phi} )}\cos \; \theta} + {( {{\cos \; \psi} + 1} ){\sin ( {\theta + \phi} )}\cos \; \theta}} \rbrack} + {2G_{xz}\sin \; {\psi sin}\; {\theta cos}\; \phi} - {2G_{yz}\sin \; {\psi cos}\; {\theta cos}\; \phi}} \rbrack}} }} }}} & (42)\end{matrix}$

Inserting explicit time dependencies, expressing products oftrigonometric functions in terms of as sums and differences of sines andcosines, and collecting terms with common frequencies gives

$\begin{matrix}{B_{u} = {{( {{{- \frac{{\overset{\_}{B}}_{x}}{2}}( {1 - {\cos \; \psi}} )} - {\frac{G_{yz}a}{4}\sin \; \psi}} ){\cos ( {\omega_{1} - \omega_{2}} )}t} - {( {{\frac{{\overset{\_}{B}}_{y}}{2}( {1 - {\cos \; \psi}} )} - {\frac{G_{xz}a}{4}\sin \; \psi}} ){\sin ( {\omega_{1} - \omega_{2}} )}t} + {{\overset{\_}{B}}_{z}\sin \; {\psi cos}\; \omega_{2}t} - {\frac{( {G_{xx} + G_{yy}} )a}{4}\sin \; \omega_{2}t} + {( {{\frac{{\overset{\_}{B}}_{x}}{2}( {1 + {\cos \; \psi}} )} - {\frac{G_{yz}a}{4}\sin \; \psi}} ){\cos ( {\omega_{1} + \omega_{2}} )}t} + {( {{\frac{{\overset{\_}{B}}_{y}}{2}( {1 + {\cos \; \psi}} )} + {\frac{G_{xz}a}{4}\sin \; \psi}} ){\sin ( {\omega_{1} + \omega_{2}} )}t} + {\frac{G_{xy}a}{4}( {1 - {\cos \; \psi}} ){\cos ( {{2\omega_{1}} - \omega_{2}} )}t} + {\frac{( {G_{yy} - G_{xx}} )a}{8}( {1 - {\cos \; \psi}} ){\sin ( {{2\omega_{1}} - \omega_{2}} )}t} - {\frac{G_{xy}a}{4}( {1 + {\cos \; \psi}} ){\cos ( {{2\omega_{1}} + \omega_{2}} )}t} + {\frac{( {G_{xx} - G_{yy}} )a}{8}( {1 + {\cos \; \psi}} ){\sin ( {{2\omega_{1}} + \omega_{2}} )}t}}} & (43)\end{matrix}$

From (43) the field and gradient components can be obtained from thefrequency components of the signal:

$\begin{matrix}{\mspace{20mu} {{{\overset{\_}{B}}_{x} = {{I( {\omega_{1} + \omega_{2}} )} - {I( {\omega_{1} - \omega_{2}} )}}},\mspace{20mu} {{\overset{\_}{B}}_{y} = {{Q( {\omega_{1} + \omega_{2}} )} - {Q( {\omega_{1} - \omega_{2}} )}}},{{\overset{\_}{B}}_{z} = {{{I( \omega_{2} )}/\sin}\; \psi}},\mspace{20mu} {G_{xy} = \frac{2\lbrack {{I( {{2\omega_{1}} - \omega_{2}} )} - {I( {{2\omega_{1}} + \omega_{2}} )}} \rbrack}{a}},\mspace{20mu} {G_{xz} = \frac{- {2\lbrack {{( {1 + {\cos \; \psi}} ){Q( {\omega_{1} - \omega_{2}} )}} + {( {1 - {\cos \; \psi}} ){Q( {\omega_{1} + \omega_{2}} )}}} \rbrack}}{a\; \sin \; \psi}},\mspace{20mu} {G_{yz} = {- \frac{2\lbrack {{( {1 + {\cos \; \psi}} ){I( {\omega_{1} - \omega_{2}} )}} + {( {1 - {\cos \; \psi}} ){I( {\omega_{1} + \omega_{2}} )}}} \rbrack}{a\; \sin \; \psi}}},{{G_{xx} - G_{yy}} = \frac{4\lbrack {{( {1 + {\cos \; \psi}} ){Q( {{2\omega_{1}} - \omega_{2}} )}} - {( {1 - {\cos \; \psi}} ){Q( {{2\omega_{1}} + \omega_{2}} )}}} \rbrack}{a\; \sin^{2}\psi}},\mspace{20mu} {G_{zz} = {{- ( {G_{xx} + G_{yy}} )} = \frac{{- 4}{Q( \omega_{2} )}}{a}}}}} & (44)\end{matrix}$

The tensor components G_(xx) and G_(yy) are determined by theexpressions for their sum and difference. Estimates of G_(xz) and G_(yz)require precise knowledge of ψ. An incorrect value of this angle willpropagate contributions from B_(x) and B_(y) into estimates of G_(xy)and G_(yz), because the signals of these gradient components have thesame frequency as the field components, unless careful calibration andreferencing is carried out, whereas gradient components G_(xy) andG_(xx)−G_(yy) are cleanly separated from field components in thefrequency domain.

Alternative ways of determining G_(xz) and G_(yz) include:

1. Calculating the differences between B_(x) and B_(y), measured usingtwo orbiting spinning sensors, separated by a baseline along Z that islong enough to reduce interactions between the sensors to a negligiblelevel, but is short compared to the distance to the nearest source ofinterest.2. Differentiating the measured values of B_(x) and B_(y) with respectto z, as the instrument is moved along the z-axis.

For the particular case ψ=0, the rotation and orbital axes are alignedand the sensor axis is confined to the orbital plane. In this case someof the frequency components vanish and the field and gradient componentsthat can be determined are

$\begin{matrix}{ \begin{matrix}\begin{matrix}\begin{matrix}{{{\overset{\_}{B}}_{x} = {I( {\omega_{1} + \omega_{2}} )}},{{\overset{\_}{B}}_{y} = {Q( {\omega_{1} + \omega_{2}} )}},} \\{{G_{xy} = {- \frac{2{I( {{2\omega_{1}} + \omega_{2}} )}}{a}}},}\end{matrix} \\{{{G_{xx} - G_{yy}} = \frac{ {4{Q( {{2\omega_{1}} + \omega_{2}} )}} \rbrack}{a}},}\end{matrix} \\{G_{zz} = {{- ( {G_{xx} + G_{yy}} )} = \frac{{- 4}{Q( \omega_{2} )}}{a}}}\end{matrix} \} ( {\psi = 0} )} & (45)\end{matrix}$

For the case ψ=90°, the rotation axis is tangential to the orbital path,with z″ parallel to −x′. In this case equation (44) simplifies to

$\begin{matrix}{ \begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{{\overset{\_}{B}}_{x} = {{I( {\omega_{1} - \omega_{2}} )} + {I( {\omega_{1} + \omega_{2}} )}}},} \\{{{\overset{\_}{B}}_{y} = {{Q( {\omega_{1} - \omega_{2}} )} + {Q( {\omega_{1} + \omega_{2}} )}}},{{\overset{\_}{B}}_{z} = {I( \omega_{2} )}},}\end{matrix} \\{{G_{xy} = {- \frac{2\lbrack {{I( {{2\omega_{1}} + \omega_{2}} )} + {I( {{2\omega_{1}} - \omega_{2}} )}} \rbrack}{a}}},}\end{matrix} \\{{G_{xz} = \frac{2\lbrack {{Q( {\omega_{1} - \omega_{2}} )} + {Q( {\omega_{1} + \omega_{2}} )}} \rbrack}{a}},}\end{matrix} \\{{G_{yz} = {- \frac{2\lbrack {{I( {\omega_{1} - \omega_{2}} )} + {I( {\omega_{1} + \omega_{2}} )}} \rbrack}{a}}},}\end{matrix} \\{{G_{xx} - G_{yy}} = \frac{4\lbrack {{Q( {{2\omega_{1}} + \omega_{2}} )} - {Q( {{2\omega_{1}} - \omega_{2}} )}} \rbrack}{a}}\end{matrix} \} ( {\psi = {90{^\circ}}} )} & (46)\end{matrix}$

These limiting cases do not provide all vector and tensor components. Anintermediate orientation of the rotation plane can provide moreinformation. For example, ψ=45° gives

$\begin{matrix} \begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{{\overset{\_}{B}}_{x} = {{I( {\omega_{1} - \omega_{2}} )} + {I( {\omega_{1} + \omega_{2}} )}}},} \\{{{\overset{\_}{B}}_{y} = {{Q( {\omega_{1} - \omega_{2}} )} + {Q( {\omega_{1} + \omega_{2}} )}}},}\end{matrix} \\{{{\overset{\_}{B}}_{z} = {\sqrt{2}{Q( \omega_{2} )}}},}\end{matrix} \\{{G_{xy} = {- \frac{2\lbrack {{I( {{2\omega_{1}} + \omega_{2}} )} + {I( {{2\omega_{1}} - \omega_{2}} )}} \rbrack}{a}}},}\end{matrix} \\{{G_{xz} = \frac{2{\sqrt{2}\lbrack {{( {1 + {1/\sqrt{2}}} ){Q( {\omega_{1} - \omega_{2}} )}} + {( {1 - {1/\sqrt{2}}} ){Q( {\omega_{1} + \omega_{2}} )}}} \rbrack}}{a}},}\end{matrix} \\{{G_{yz} = {- \frac{2{\sqrt{2}\lbrack {{( {1 + {1/\sqrt{2}}} ){I( {\omega_{1} - \omega_{2}} )}} + {( {1 - {1/\sqrt{2}}} ){Q( {\omega_{1} + \omega_{2}} )}}} \rbrack}}{a}}},}\end{matrix} \\{{{G_{xx} - G_{yy}} = \frac{8\begin{bmatrix}{{( {1 - {1/\sqrt{2}}} ){Q( {{2\omega_{1}} + \omega_{2}} )}} -} \\{( {1 + {1/\sqrt{2}}} ){Q( {{2\omega_{1}} - \omega_{2}} )}}\end{bmatrix}}{a}},} \\{G_{zz} = {{- ( {G_{xx} + G_{yy}} )} = \frac{4\sqrt{2}{Q( \omega_{2} )}}{a}}}\end{matrix} \} & (47) \\{\mspace{20mu} ( {\psi = {45{^\circ}}} )} & \;\end{matrix}$

The sensor trajectory shown in FIGS. 8A to 8E is the same as thetrajectory shown in FIGS. 2A to 2D, but the sensitive axis u is rotatingabout an inclined axis relative to X′. A third reference frameX″=(x″,y″,z″), which is different from the third reference frame X″ ofFIGS. 7A to 7E, is inclined and fixed with respect to X′. The inclinedframe X″ is rotated about x′ by an angle ψ such that x″=x, there is anangle ψ between z″ and z′ and there is an angle ψ between y″ and y′. Theangle ψ is defined as positive as measured between z′ and z″ from z′towards y′.

The sensitive axis u is located at the X″ origin O″=O′r₁ and rotates inthe x″y″-plane about z″ (and perpendicular to z″) with angular velocityω₂ and rotation angle φ (φ=0 when u is parallel to x″, φ is defined aspositive as measured between x″ and u from x″ towards y″, and φ=ω₂t,where t=time).

The field components at an instantaneous measurement point r₁(x,y) are

B(x,y)= B +(a/2)(G _(xx) cos θ+G _(xy) sin θ){circumflex over(x)}+(a/2)(G _(xy) cos θ+G _(yy) sin θ)ŷ+(a/2)(G _(xz) cos θ+G _(yz) sinθ){circumflex over (z)}  (48)

The field component measured by the sensor 110 is given by:

B _(u) =B _(x″) cos φ+B _(y″) sin φ=B _(x′) cos φ+B _(y″) sin φ,  (49)

where

B _(y″) =B _(y′) cos ψ−B _(z′) sin ψ=B _(y′) cos ψ−B _(z) sin ψ,  (50)

B _(x′) =B _(x) cos θ+B _(y) sin θ,  (51)

B _(y′) =−B _(x) sin θ+B _(y) cos θ.  (52)

Substituting (50)-(52) into (49) gives

$\begin{matrix}{B_{u} = {{{\lbrack {{( {{B_{x}\cos \; \theta} + {B_{y}\sin \; \theta}} )\cos \; \psi} - {B_{z}\sin \; \psi}} \rbrack \cos \; \phi} + {( {{{- B_{x}}\sin \; \theta} + {B_{y}\cos \; \theta}} )\sin \; \phi}} = {{{B_{x}\lbrack {{\cos \; {\psi cos}\; {\theta cos}\; \phi} - {\sin \; {\theta sin}\; \phi}} \rbrack} + {B_{y}\lbrack {{\cos \; {\psi cos}\; {\phi sin}\; \theta} + {\sin \; {\phi cos}\; \theta}} \rbrack} - {B_{z}\sin \; {\psi cos}\; \phi}} = {{B_{x}\lbrack {{\frac{\cos \; \psi}{2}\lbrack {{\cos ( {\phi - \theta} )} + {\cos ( {\phi + \theta} )}} \rbrack} - {\frac{1}{2}\lbrack {{\cos ( {\phi - \theta} )} - {\cos ( {\phi + \theta} )}} \rbrack}} \rbrack} + {B_{y}\lbrack {{\frac{\cos \; \psi}{2}\lbrack {{- {\sin ( {\phi - \theta} )}} + {\sin ( {\phi + \theta} )}} \rbrack} + {\frac{1}{2}\lbrack {{\sin ( {\phi - \theta} )} + {\sin ( {\phi + \theta} )}} \rbrack}} \rbrack} - {B_{z}\sin \; {\psi cos}\; \theta}}}}} & (53)\end{matrix}$

Substituting (48) into (53) and rearranging gives

$\begin{matrix}{B_{u} = {{\frac{{\overset{\_}{B}}_{x}}{2}\lbrack {{( {{\cos \; \psi} - 1} ){\cos ( {\phi - \theta} )}} + {( {{\cos \; \psi} + 1} ){\cos ( {\phi + \theta} )}}} \rbrack} + {\frac{{\overset{\_}{B}}_{y}}{2}\lbrack {{( {{\cos \; \psi} - 1} ){\sin ( {\phi - \theta} )}} + {( {{\cos \; \psi} + 1} ){\sin ( {\phi + \theta} )}}} \rbrack} - {{\overset{\_}{B}}_{z}\sin \; {\psi cos}\; \phi} + {\frac{a}{4}\lbrack {{G_{xx}\lbrack {{( {{\cos \; \psi} - 1} ){\cos ( {\phi - \theta} )}\cos \; \theta} + {( {{\cos \; \psi} + 1} ){\cos ( {\phi + \theta} )}\cos \; \theta}} \rbrack} + {G_{xy}\lbrack {{( {{\cos \; \psi} - 1} ){\cos ( {\phi - \theta} )}\sin \; \theta} + {( {{\cos \; \psi} + 1} ){\cos ( {\phi + \theta} )}\sin \; \theta}} \rbrack} + {G_{xy}\lbrack {{( {{\cos \; \psi} - 1} ){\sin ( {\phi - \theta} )}\cos \; \theta} + {( {{\cos \; \psi} + 1} ){\sin ( {\phi + \theta} )}\cos \; \theta}} \rbrack} + {G_{yy}\lbrack {{( {{\cos \; \psi} - 1} ){\sin ( {\phi - \theta} )}\sin \; \theta} + {( {{\cos \; \psi} + 1} ){\sin ( {\phi + \theta} )}\sin \; \theta}} \rbrack} - {G_{xz}\sin \; {\psi \lbrack {{\cos ( {\phi - \theta} )} + {\cos ( {\phi + \theta} )}} \rbrack}} + {G_{yz}\sin \; {\psi( {\sin ( {\phi - \theta} )} \rbrack}}} \rbrack}}} & (54)\end{matrix}$

Inserting explicit time dependencies, expressing products oftrigonometric functions in terms of as sums and differences of sines andcosines, and collecting terms with common frequencies gives

$\begin{matrix}{B_{u} = {{( {{{- {\overset{\_}{B}}_{z}}\sin \; \psi} + {\frac{( {G_{xx} + G_{yy}} )a}{8}\cos \; \psi}} ){\cos ( {\omega_{2}t} )}( {{{- \frac{{\overset{\_}{B}}_{x}}{2}}( {1 - {\cos \; \psi}} )} - {\frac{G_{xz}a}{8}\sin \; \psi}} ){\cos ( {( {\omega_{2} - \omega_{1}} )t} )}} + {( {{\frac{{\overset{\_}{B}}_{y}}{2}( {1 - {\cos \; \psi}} )} + {\frac{G_{yz}a}{8}\sin \; \psi}} ){\sin ( {( {\omega_{2} - \omega_{1}} )t} )}} + {( {{\frac{{\overset{\_}{B}}_{x}}{2}( {1 + {\cos \; \psi}} )} - {\frac{G_{xz}a}{8}\sin \; \psi}} ){\cos ( {( {\omega_{2} + \omega_{1}} )t} )}} + {( {{\frac{{\overset{\_}{B}}_{y}}{2}( {1 + {\cos \; \psi}} )} - {\frac{G_{yz}a}{8}\sin \; \psi}} ){\sin ( {( {\omega_{2} + \omega_{1}} )t} )}} + {\frac{( {G_{yy} - G_{xx}} )a}{8}( {1 - {\cos \; \psi}} ){\cos ( {( {{2\omega_{2}} - \omega_{1}} )t} )}} + {\frac{G_{xy}a}{4}( {1 - {\cos \; \psi}} ){\sin ( {( {\omega_{2} - {2\omega_{1}}} )t} )}} + {\frac{( {G_{xx} - G_{yy}} )a}{8}( {1 + {\cos \; \psi}} ){\cos ( {( {{2\omega_{2}} + \omega_{1}} )t} )}} + {\frac{G_{xy}a}{4}( {1 + {\cos \; \psi}} ){\sin ( {( {\omega_{2} + {2\omega_{1}}} )t} )}}}} & (55)\end{matrix}$

From (55) the field and gradient components can be obtained from thefrequency components of the signal:

$\begin{matrix}{{{\overset{\_}{B}}_{x} = {{I( {\omega_{2} + \omega_{1}} )} - {I( {\omega_{2} - \omega_{1}} )}}},{{\overset{\_}{B}}_{y} = {{Q( {\omega_{2} + \omega_{1}} )} + {Q( {\omega_{2} - \omega_{1}} )}}},{{{\overset{\_}{B}}_{z} + {\frac{G_{zz}a}{8}\cos \; \psi}} = {{{- {I( \omega_{2} )}}/\sin}\; \psi}},{G_{xy} = \frac{2\lbrack {{Q( {{2\omega_{2}} - \omega_{1}} )} + {Q( {{2\omega_{2}} + \omega_{1}} )}} \rbrack}{a}},{G_{{xz}\;} = \frac{- {4\lbrack {{I( {\omega_{2} - \omega_{1}} )} + {I( {\omega_{2} + \omega_{1}} )}} \rbrack}}{a\; \sin \; \psi}},{G_{yz} = \frac{4\lbrack {{Q( {\omega_{2} - \omega_{1}} )} - {Q( {\omega_{2} + \omega_{1}} )}} \rbrack}{a\; \sin \; \psi}},{{G_{xx} - G_{yy}} = \frac{4\lbrack {{I( {\omega_{2} + {2\omega_{1}}} )} - {I( {\omega_{2} - {2\omega_{1}}} )}} \rbrack}{a}}} & (56)\end{matrix}$

The ω₁ frequency component will normally be dominated by B_(z). It isdifficult to separate B_(z) and G_(zz) from measurements at a singlelocation. Measurements made at closely spaced points as the sensor movesalong the z axis allow independent estimates of B_(z) and G_(zz):

$\begin{matrix}{{{- \frac{\lbrack {{I_{\omega_{2}}( {z_{0} + {\Delta \; {z/2}}} )} - {I_{\omega_{2}}( {z_{0} - {\Delta \; {z/2}}} )}} \rbrack}{\Delta \; z\; \sin \; \psi}} = {{{G_{zz}( z_{0} )} + \frac{{G_{zzz}( z_{0} )}a\; \cot \; \psi}{8}} \approx {G_{zz}( z_{0} )}}},{{\frac{- 1}{\sin \; \psi}\lbrack {{I_{\omega_{2}}( z_{0} )} - \frac{\lbrack {{I_{\omega_{2}}( {z_{0} + {\Delta \; {z/2}}} )} - {I_{\omega_{2}}( {z_{0} - {\Delta \; {z/2}}} )}} \rbrack a\; \cot \; \psi}{8\Delta \; z}} \rbrack} = {{{{\overset{\_}{B}}_{z}( z_{0} )} - \frac{{G_{zzz}( z_{0} )}a^{2}\cot^{2}\psi}{8}} \approx {{\overset{\_}{B}}_{z}( z_{0} )}}}} & (57)\end{matrix}$

The approximate equalities on the right-hand side of the expressions forG_(zz) and B_(z) are exact if the gradient is uniform, and are goodapproximations if the distance to the nearest magnetic source is atleast several times the orbital diameter a. The tensor components G_(xx)and G_(yy) are determined by the expressions for their sum, which is−G_(zz), and difference. Estimates of G_(xz) and G_(yz) require preciseknowledge of ψ. An incorrect value of this angle will propagatecontributions from B_(x) and B_(y) into estimates of G_(xz) and G_(yz),because the signals of these gradient components have the same frequencyas the field components, unless careful calibration and referencing iscarried out, whereas gradient components G_(xy) and G_(xx)−G_(yy) arecleanly separated from field components in the frequency domain.

FIGS. 9A to 9D show a sensor arrangement and trajectories for twosensors S1 and S2 according to some embodiments. The two sensors S1 andS2 follow parallel circular trajectories with the trajectory and sensorrotation of each sensor being similar to that described in relation toFIGS. 8A to 8E, and planes of each trajectory are separated by a certaindistance in the z-direction of the instrument reference frame.

The sensors S1 and S2 are mounted at opposite ends of a common rotationaxis (collinear with z′_(S1) and z′_(S2)) which precesses about amidpoint between the two sensors. The precession axis is inclinedrelative to the instrument z-axis by an angle ψ, and precesses aroundthe instrument z-axis at an angular velocity of ω₁.

In some embodiments, the two sensors S1 and S2 may be separated by alarge enough distance that magnetic interference between is negligiblecompared to the measured signal. In some embodiments, the sensors S1 andS2 may be inclined with respect to each other in their respectivex″y″-planes and sensitive axes u₁ and u₂ may be offset by a differencein rotation angle Δφ in order to further reduce or mitigate magneticinterference between the sensors. The difference in rotation angle Δφmay be any suitable angle between 0° and 180°, such as 90°, for example.In other embodiments, the sensors S1 and S2 may be fixed in parallelrelative to each other with Δφ=0°.

The sensitive axes of the sensors S1 and S2 may be denoted by u₁ and u₂respectively, and the measured signals from each sensor may be combinedto determine the components of the gradient tensor in the z-direction.

In the plane of the orbit of S1, the field at the instantaneous position(x, y)=(a/2) (sin θ, −cos θ) of the sensor S1 is then

B _(S1)(x,y)= B _(S1)+(a/2)(G _(xx) sin θ−G _(xy) cos θ){circumflex over(x)}+(a/2)(G _(xy) sin θ−G _(yy) cos θ)ŷ+(a/2)(G _(xz) sin θ−G _(yz) cosθ){circumflex over (z)}.  (58)

The signal from the spinning orbiting sensor S1 is

B _(u) ₁ =B _(y) _(S1) _(″) cos φ+B _(x) _(S1) _(″) sin φ,  (59)

where

B _(y) _(S1) _(″) =B _(y) _(S1) cos ψ−B _(z) sin ψ,  (60)

B _(x) _(S1) _(″) =B _(x) cos θ+B _(y) sin θ,  (61)

B _(y) _(S1) =−B _(x) sin θ+B _(y) cos θ.  (62)

Substituting (60)-(62) into (59) gives

$\begin{matrix}{B_{u_{1}} = {{{\lbrack {{( {{{- B_{x}}\sin \; \theta} + {B_{y}\cos \; \theta}} )\cos \; \psi} - {B_{z}\sin \; \psi}} \rbrack \cos \; \phi} + {( {{B_{x}\cos \; \theta} + {B_{y}\sin \; \theta}} )\sin \; \phi}} = {{{B_{x}\lbrack {{{- \cos}\; \psi \; \sin \; {\theta cos}\; \phi} + {\cos \; {\theta sin}\; \phi}} \rbrack} + {B_{y}\lbrack {{\cos \; {\psi cos}\; {\theta cos}\; \phi} + {\sin \; {\theta sin}\; \phi}} \rbrack} - {B_{z}\sin \; {\psi cos}\; \phi}} = {{B_{x}\lbrack {{- {\frac{\cos \; \psi}{2}\lbrack {{\sin ( {\theta - \phi} )} + {\sin ( {\theta + \phi} )}} \rbrack}} - {\frac{1}{2}\lbrack {{\sin ( {\theta - \phi} )} - {\sin ( {\theta + \phi} )}} \rbrack}} \rbrack} + {B_{y}\lbrack {{\frac{\cos \; \psi}{2}\lbrack {{\cos ( {\theta - \phi} )} + {\cos \; ( {\theta + \phi} )}} \rbrack} + {\frac{1}{2}\lbrack {{\cos ( {\theta - \phi} )} - {\cos ( {\theta + \phi} )}} \rbrack}} \rbrack} - {B_{z}\sin \; {\psi cos}\; {\phi.}}}}}} & (63)\end{matrix}$

Substituting (58) into (63) and rearranging gives

$\begin{matrix}{B_{u_{1}} = {{\frac{{\overset{\_}{B}}_{x}}{2}\lbrack {{( {{\cos \; \psi} - 1} ){\cos ( {\theta - \phi} )}} + {( {{\cos \; \psi} + 1} ){\cos ( {\theta + \phi} )}}} \rbrack} + {\frac{{\overset{\_}{B}}_{y}}{2}\lbrack {{( {{\cos \; \psi} - 1} ){\sin ( {\theta - \phi} )}} + {( {{\cos \; \psi} + 1} ){\sin ( {\theta + \phi} )}}} \rbrack} - {{\overset{\_}{B}}_{z}\sin \; \psi \; \cos \; \theta} + {{\frac{b}{4}\lbrack {{G_{xx}\lbrack {{( {{\cos \; \psi} - 1} ){\cos ( {\theta - \phi} )}\cos \; \phi} + {( {{\cos \; \psi} + 1} ){\cos ( {\theta + \phi} )}\cos \; \phi}} \rbrack} + {G_{xy}\lbrack {{( {{\cos \; \psi} - 1} ){\cos ( {\theta - \phi} )}\sin \; \phi} + {( {{\cos \; \psi} + 1} ){\cos ( {\theta + \phi} )}\sin \; \phi}} \rbrack} + {G_{xy}\lbrack {{( {{\cos \; \psi} - 1} ){\sin ( {\theta - \phi} )}\; \cos \; \phi} + {( {{\cos \; \psi} + 1} ){\sin ( {\theta + \phi} )}\cos \; \phi}} \rbrack} + {G_{yy}\lbrack {{( {{\cos \; \psi} - 1} ){\sin ( {\theta - \phi} )}\sin \; \phi} + {( {{\cos \; \psi} + 1} ){\sin ( {\theta + \phi} )}\sin \; \phi}} \rbrack} - {G_{xz}\sin \; {\psi \lbrack {{\cos ( {\theta - \phi} )} + {\cos ( {\theta + \phi} )}} \rbrack}} + {G_{yz}\sin \; {\psi \lbrack {{\sin ( {\theta - \phi} )} - {\sin ( {\theta + \phi} )}} \rbrack}}} \rbrack}.}}} & (64)\end{matrix}$

Inserting explicit time dependencies, expressing products oftrigonometric functions in terms of as sums and differences of sines andcosines, and collecting terms with common frequencies gives

$\begin{matrix}{B_{u_{1}} = {{( {{{- {\overset{\_}{B}}_{z}}\sin \; \psi} - {\frac{( {G_{xx} + G_{yy}} )a}{4}\cos \; \psi}} )\cos \; \omega_{2}{t( {{{- \frac{{\overset{\_}{B}}_{x}}{2}}( {1 + {\cos \; \psi}} )} - {\frac{G_{xz}a}{4}\sin \; \psi}} )}{\sin ( {\omega_{1} - \omega_{2}} )}t} + {( {{\frac{{\overset{\_}{B}}_{y}}{2}( {1 + {\cos \; \psi}} )} + {\frac{G_{yz}a}{4}\sin \; \psi}} ){\cos ( {\omega_{1} - \omega_{2}} )}t} + {( {{\frac{{\overset{\_}{B}}_{x}}{2}( {1 - {\cos \; \psi}} )} - {\frac{G_{{xz}\;}a}{4}\sin \; \psi}} ){\sin ( {\omega_{1} + \omega_{2}} )}t} - {( {{\frac{{\overset{\_}{B}}_{y}}{2}( {1 - {\cos \; \psi}} )} - {\frac{G_{yz}a}{4}\sin \; \psi}} ){\cos ( {\omega_{1} + \omega_{2}} )}t} + {\frac{( {G_{xx} - G_{yy}} )a}{8}( {1 + {\cos \; \psi}} ){\cos ( {{2\omega_{1}} - \omega_{2}} )}t} + {\frac{G_{xy}a}{4}( {1 + {\cos \; \psi}} ){\sin ( {{2\omega_{1}} - \omega_{2}} )}t} - {\frac{( {G_{xx} - G_{yy}} )a}{8}( {1 - {\cos \; \psi}} ){\cos ( {{2\omega_{1}} + \omega_{2}} )}t} - {\frac{G_{xy}a}{4}( {1 - {\cos \; \psi}} ){\sin ( {{2\omega_{1}} + \omega_{2}} )}{t.}}}} & (65)\end{matrix}$

From (65) the field and gradient components can be obtained from thefrequency components of the signal:

$\begin{matrix}{{{\overset{\_}{B}}_{x} = {{Q( {\omega_{1} + \omega_{2}} )} - {Q( {\omega_{1} - \omega_{2}} )}}},{{\overset{\_}{B}}_{y} = {{I( {\omega_{1} - \omega_{2}} )} - {I( {\omega_{1} + \omega_{2}} )}}},{{{\overset{\_}{B}}_{z} - {\frac{G_{zz}a}{8}\cot \; \psi}} = {{{- {I( \omega_{2} )}}/\sin}\; \psi}},{G_{xy} = \frac{2\lbrack {{Q( {{2\omega_{1}} - \omega_{2}} )} - {Q( {{2\omega_{1}} + \omega_{2}} )}} \rbrack}{a}},{G_{xz} = \frac{- {2\lbrack {{( {1 - {\cos \; \psi}} ){Q( {\omega_{1} - \omega_{2}} )}} + {( {1 + {\cos \; \psi}} ){Q( {\omega_{1} + \omega_{2}} )}}} \rbrack}}{a\; \sin \; \psi}},{G_{yz} = \frac{2\lbrack {{( {1 - {\cos \; \psi}} ){I( {\omega_{1} - \omega_{2}} )}} + {( {1 + {\cos \; \psi}} ){I( {\omega_{1} + \omega_{2}} )}}} \rbrack}{a\; \sin \; \psi}},{{G_{xx} - G_{yy}} = {\frac{4\lbrack {{I( {{2\omega_{1}} - \omega_{2}} )} - {I( {{2\omega_{1}} + \omega_{2}} )}} \rbrack}{a}.}}} & (66)\end{matrix}$

The ω₂ frequency component will normally be dominated by B_(z). It maybe difficult to separate B_(z) and G_(zz) completely from measurementsof a single sensor precessing in a single orbital plane. Measurementsmade at closely spaced points as the sensor moves along the z axis allowindependent estimates of B_(z) and G_(zz):

$\begin{matrix}{{{- \frac{\lbrack {{I_{\omega_{1}}( {z_{0} + {\Delta \; {z/2}}} )} - {I_{\omega_{1}}( {z_{0} - {\Delta \; {z/2}}} )}} \rbrack}{\Delta \; z\; \sin \; \psi}} = {{{G_{zz}( z_{0} )} + \frac{{G_{zzz}( z_{0} )}b\; \cot \; \psi}{8}} \approx {G_{zz}( z_{0} )}}},{{\frac{- 1}{\sin \; \psi}\lbrack {{I_{\omega_{1}}( z_{0} )} - \frac{\lbrack {{I_{\omega_{1}}( {z_{0} + {\Delta \; {z/2}}} )} - {I_{\omega_{1}}( {z_{0} - {\Delta \; {z/2}}} )}} \rbrack b\; \cot \; \psi}{8\Delta \; z}} \rbrack} = {{{{\overset{\_}{B}}_{z}( z_{0} )} - \frac{{G_{zzz}( z_{0} )}b^{2}\cot^{2}\psi}{8}} \approx {{{\overset{\_}{B}}_{z}( z_{0} )}.}}}} & (67)\end{matrix}$

The tensor components G_(xx) and G_(yy) are determined by theexpressions for their sum, which is −G_(zz), and difference. Estimatesof G_(xz) and G_(yz) require precise knowledge of ψ. An incorrect valueof this angle will propagate contributions from B_(x) and B_(y) intoestimates of G_(xz) and G_(yz), because the signals of these gradientcomponents have the same frequency as the field components, unlesscareful calibration and referencing is carried out, whereas gradientcomponents G_(xy) and G_(xx)−G_(yy) are cleanly separated from fieldcomponents in the frequency domain.

For the two sensor embodiment shown in FIGS. 9A to 9D, the expressionsfor the field and gradient components of sensor S2 have the same form asthose for S1, but apply to fields in the orbital plane of S2, for whichz=+z₀/2, whereas the orbital plane of S1 is at z=−z₀/2, referred to anorigin of Z at the midway between the two orbital planes. Theinstantaneous position of the sensor S2 within its orbital plane is (x,y)=(a/2)(−sin θ, cos θ), so for S2:

B _(S2)(x,y)= B _(S2)(z=z ₀/2)−(a/2)(G _(xx) sin θ−G _(xy) cosθ){circumflex over (x)}−(a/2)(G _(xy) sin θ−G _(yy) cos θ)ŷ−(a/2)(G_(xz) sin θ−G _(yz) cos θ){circumflex over (z)}.  (68)

Applying the same analysis to equation (68) and combining with equation(58), the field and gradient components at the midpoint of the twosensor configuration are given by

$\begin{matrix}{{{\overset{\_}{B}}_{x} = {\lbrack {{Q_{1}( {\omega_{1} + \omega_{2}} )} + {Q_{2}( {\omega_{1} + \omega_{2}} )} - {Q( {\omega_{1} - \omega_{2}} )} - {Q_{2}( {\omega_{1} - \omega_{2}} )}} \rbrack/2}},{{\overset{\_}{B}}_{y} = {\lbrack {{I_{1}( {\omega_{1} - \omega_{2}} )} + {I_{2}( {\omega_{1} - \omega_{2}} )} - {I_{2}( {\omega_{1} + \omega_{2}} )} - {I_{2}( {\omega_{1} + \omega_{2}} )}} \rbrack/2}},\mspace{20mu} {{\overset{\_}{B}}_{z} = {{{- \lbrack {{I_{1}( \omega_{2} )} + {I_{2}( \omega_{2} )}} \rbrack}/2}\; \sin \; \psi}},} & (69) \\{\mspace{20mu} {{{\overset{\_}{G}}_{xy} = \frac{\begin{bmatrix}{{Q_{1}( {{2\omega_{1}} - \omega_{2}} )} + {Q_{2}( {{2\omega_{1}} - \omega_{2}} )} -} \\{{Q_{1}( {{2\omega_{1}} + \omega_{2}} )} - {Q_{2}( {{2\omega_{1}} + \omega_{2}} )}}\end{bmatrix}}{a}},}} & \; \\{{\overset{\_}{G}}_{xz} = {\quad{{\lbrack {{Q_{2}( {\omega_{1} + \omega_{2}} )} - {Q_{2}( {\omega_{1} - \omega_{2}} )} - {Q_{1}( {\omega_{1} + \omega_{2}} )} + {Q_{1}( {\omega_{1} - \omega_{2}} )}} \rbrack/z_{0}},}}} & \; \\{{{\overset{\_}{G}}_{yz} = {\lbrack {{I_{2}( {\omega_{1} - \omega_{2}} )} - {I_{2}( {\omega_{1} + \omega_{2}} )} - {I_{1}( {\omega_{1} - \omega_{2}} )} + {I_{1}( {\omega_{1} + \omega_{2}} )}} \rbrack/z_{0}}},\mspace{20mu} {{{\overset{\_}{G}}_{xx} - {\overset{\_}{G}}_{yy}} = \frac{2\begin{bmatrix}{{I_{1}( {{2\omega_{1}} - \omega_{2}} )} + {I_{2}( {{2\omega_{1}} - \omega_{2}} )} -} \\{{I_{1}( {{2\omega_{1}} + \omega_{2}} )} - {I_{2}( {{2\omega_{1}} + \omega_{2}} )}}\end{bmatrix}}{a}},\mspace{20mu} {{\overset{\_}{G}}_{zz} = {{- ( {{\overset{\_}{G}}_{xx} + {\overset{\_}{G}}_{yy}} )} = {{2\lbrack {{I_{1}( \omega_{2} )} - {I_{2}( \omega_{2} )}} \rbrack}/{( {{a\; \cos \; \psi} - {2z_{0}\sin \; \psi}} ).}}}}} & \;\end{matrix}$

In some embodiments, the X″ frame may be inclined with respect to X androtated about z′ instead of x′ or y′. In other embodiments, X″ may beinclined with respect to X and rotated about x′ and y′, or y′ and z′, orz′ and x′, or x′ and y′ and z′. In various alternate embodiments, thesensitive axis u of the sensor 110 may rotate about any of the axes x′,y′, z′, x″, y″ or z″ of any of the previously described embodiments andmay be directed perpendicular to the axis of rotation, or inclined withrespect to the axis of rotation.

In the sensor trajectory shown in FIGS. 10A to 10D, the X′ frame orbitsthe origin O at r₁=O′ as described in relation to FIGS. 2A to 2D.However, a third reference frame X″, which is different from the thirdreference frames of FIGS. 4A to 4D and 5A to 5D, is spaced from the X′frame and orbits the origin O′ at r₂=O″ in a plane of rotation which istangential to the trajectory of O′ with angular velocity ω₂ and rotationangle φ (φ=0 when x″ is parallel to x′, φ is defined as positive asmeasured between x′ and x″ from x′ towards z′, and φ=ω₂t, where t=time).

The path described by r₂ is a circular orbit about origin O′ in thex′z′-plane at a distance |r₂|=b/2 from the origin O′, and the trajectoryof the X″ frame is such that x″ is directed in the direction of motion,tangential to the orbit about origin O′; z″ is directed radially inwardtowards origin O; and y″ is directed parallel to and spaced from y′. Thesensor 110 is positioned at the origin O″ and the sensitive axis u isfixed relative to frame X″ and directed along the x″-axis, but may bedirected in various other directions in other embodiments.

The position of the sensor r=r₁+r₂=(x,y,z) at time t is:

$\begin{matrix}{{x = {{\frac{a}{2}\cos \; \omega_{1}t} - {\frac{b}{2}\sin \; \omega_{1}t\; \sin \; \omega_{2}t}}},{y = {{\frac{a}{2}\sin \; \omega_{1}} + {\frac{b}{2}\cos \; \omega_{1}t\; \sin \; \omega_{2}t}}},{z = {\frac{b}{2}\cos \; \omega_{2}t}}} & (70)\end{matrix}$

The magnetic field vector at the sensor is:

$\begin{matrix}\begin{matrix}{{B( {x,y,z} )} = {\overset{\_}{B} + {G \cdot r}}} \\{= {\overset{\_}{B} + {( {{G_{xx}x} + {G_{xy}y} + {G_{xz}z}} )\hat{x}} + {( {{G_{xy}x} + {G_{yy}y} + {G_{yz}z}} )\hat{y}} +}} \\{{( {{G_{xz}x} + {G_{yz}y} + {G_{zz}z}} )\hat{z}}}\end{matrix} & (71)\end{matrix}$

The field component measured by the sensor is:

$\begin{matrix}\begin{matrix}{B_{u} = {B \cdot \hat{u}}} \\{= {B \cdot \lbrack {{{- \hat{x}}\; \cos \; \omega_{1}t\; \cos \; \omega_{2}t} - {\hat{y}\; \sin \; \omega_{1}t\; \cos \; \omega_{2}t} - {\hat{z}\; \sin \; \omega_{2}t}} \rbrack}} \\{= {{{- B_{x}}\cos \; \omega_{1}t\; \cos \; \omega_{2}t} - {B_{y}\sin \; \omega_{1}t\; \cos \; \omega_{2\;}t} - {B_{z}\sin \; \omega_{2}t}}}\end{matrix} & (72)\end{matrix}$

Combining (70)-(72) and converting products of trigonometric functionsinto individual frequency components (sums and differences of multiplesof ω₁ and ω₂) gives:

$\begin{matrix}{B_{u} = {{- {{\overset{\_}{B}}_{x}\lbrack {{\cos ( {( {\omega_{1} - \omega_{2}} )t} )} + {\cos ( {( {\omega_{1} + \omega_{2}} )t} )}} \rbrack}} + {{\overset{\_}{B}}_{y}\lbrack {{- {\sin ( {( {\omega_{1} - \omega_{2}} )t} )}} - {{\sin ( ( {\omega_{1} + \omega_{2}} ) )}t}} \rbrack} - {B_{z}\sin \; \omega_{2}t} - {\frac{G_{xx}a}{8}\lbrack {{- {\sin ( {( {{2\omega_{1}} - \omega_{2}} )t} )}} - {\sin ( {( {{2\omega_{1}} + \omega_{2}} )t} )}} \rbrack} + {\frac{G_{xx}b}{16}\lbrack {{2{\sin ( {2\omega_{2}t} )}} - {\sin ( {( {{2\omega_{1}} - {2\omega_{2}}} )t} )} + {\sin ( {( {{2\omega_{1}} + {2\omega_{2}}} )t} )}} \rbrack} + {\frac{G_{xy}a}{4}\lbrack {{- {\cos ( {( {{2\omega_{1}} - \omega_{2\;}} )t} )}} - {\cos ( {( {{2\omega_{1}} + \omega_{2}} )t} )}} \rbrack} - {\frac{G_{xy}b}{8}\lbrack {{- {\cos ( {( {{2\omega_{1}} - {2\omega_{2}}} )t} )}} + {\cos ( {( {{2\omega_{1}} + {2\omega_{2}}} )t} )}} \rbrack} + {\frac{G_{xz}a}{4}\lbrack {{\cos ( {( {\omega_{1} - \omega_{2}} )t} )} - {\cos ( {( {\omega_{1} + \omega_{2}} )t} )}} \rbrack} - {\frac{G_{xz}b}{4}\lbrack {{\cos ( {( {\omega_{1} - {2\omega_{2}}} )t} )} + {\cos ( {( {\omega_{1} + {2\omega_{2}}} )t} )}} \rbrack} + {\frac{G_{yy}a}{8}\lbrack {{- {\sin ( {( {{2\omega_{1}} - \omega_{2}} )t} )}} - {\sin ( {( {{2\omega_{1}} + \omega_{2}} )t} )}} \rbrack} + {\frac{G_{yy}b}{16}\lbrack {{2{\sin ( {2\omega_{2}t} )}} + {\sin ( {( {{2\omega_{1}} - {2\omega_{2}}} )t} )} - {\sin ( {( {{2\omega_{1}} + {2\omega_{2}}} )t} )}} \rbrack} - {\frac{G_{yz}a}{4}\lbrack {{{- {\sin ( ( {\omega_{1} - \omega_{2}} ) )}}t} + {\sin ( {( {\omega_{1} + \omega_{2}} )t} )}} \rbrack} + {\frac{G_{yz}b}{2}\lbrack {{- {\sin ( {( {\omega_{1} - {2\omega_{2}}} )t} )}} - {\sin ( {( {\omega_{1} + {2\omega_{2}}} )t} )}} \rbrack} - {\frac{3G_{zz}b}{8}{\sin ( {2\omega_{2}t} )}}}} & (73)\end{matrix}$

Equation (73) shows that all vector and tensor components can be derivedfrom the Fourier components of the output. If the rotation rates arechosen such that ω₁=5ω₂, for example, then the frequency componentspresent in the signal are f₂, 2f₂, 3f₂, 4f₂, 6f₂, 7f₂, 8f₂, 9f₂, 11f₂,and 12f₂, where f₂=ω₂/2π. The vector and tensor components can beobtained from the discrete frequency components as follows:

$\begin{matrix}{\mspace{20mu} {{{\overset{\_}{B}}_{x} = {- \frac{{Q( {\omega_{1} - \omega_{2}} )} + {Q( {\omega_{1} + \omega_{2}} )}}{2}}},\mspace{20mu} {{\overset{\_}{B}}_{y} = \frac{{I( {\omega_{1} - \omega_{2}} )} + {I( {\omega_{1} + \omega_{2}} )}}{2}},{{\overset{\_}{B}}_{z} = {- {Q( \omega_{2} )}}},{{G_{xx} - G_{yy}} = {{- \frac{8\lbrack {{Q( {{2\omega_{1}} - \omega_{2}} )} + {Q( {{2\omega_{1}} + \omega_{2}} )}} \rbrack}{a}} = \frac{8\lbrack {{Q( {{2\omega_{1}} - {2\omega_{2}}} )} + {Q( {{2\omega_{1}} + {2\omega_{2}}} )}} \rbrack}{b}}},\mspace{20mu} {G_{xy} = \frac{4\lbrack {{I( {{2\omega_{1}} + {2\omega_{2}}} )} - {I( {{2\omega_{1}} - {2\omega_{2}}} )}} \rbrack}{b}},\mspace{20mu} {G_{xz} = {- \frac{2\lbrack {{Q( {\omega_{1} - {2\omega_{2}}} )} + {Q( {\omega_{1} + {2\omega_{2}}} )}} \rbrack}{b}}},\mspace{20mu} {G_{yz} = \frac{\lbrack {{I( {\omega_{1} - {2\omega_{2}}} )} + {I( {\omega_{1} + {2\omega_{2}}} )}} \rbrack}{b}},\mspace{20mu} {G_{zz} = {{- ( {G_{xx} + G_{yy}} )} = {- \frac{8{Q( {2\omega_{2}} )}}{3b}}}},\mspace{20mu} {G_{xx} = \frac{8\lbrack {{Q( {2\omega_{2}} )} + {3{Q( {{2\omega_{1}} - {2\omega_{2}}} )}} + {3{Q( {{2\omega_{1}} + {2\omega_{2}}} )}}} \rbrack}{6b}},\mspace{20mu} {G_{yy} = \frac{8\lbrack {{Q( {2\omega_{2}} )} - {3{Q( {{2\omega_{1}} - {2\omega_{2}}} )}} - {3{Q( {{2\omega_{1}} + {2\omega_{2}}} )}}} \rbrack}{6b}}}} & (74)\end{matrix}$

The angular velocities ω₁ and ω₂ should be commensurate andphase-locked, with their ratio chosen to ensure that different frequencycomponents in the signal given by (73) do not coincide. As mentionedabove, ω₁=5ω₂ fulfils this requirement. Many other choices, e.g.ω₁=2.5ω₂, or ω₁=7ω₂, are also suitable and in some embodiments, ω₂ maybe larger than ω₁. In other embodiments, the angular velocities ω₁ andω₂ may not be commensurate and phase locked provided the phase of eachrotation is measured independently. The angular velocities ω₁ and ω₂should be configured to allow clean separation of different frequencycomponents in the signal. Commensurability may be desirable for digitalimplementation using a discrete Fourier transform, for example, as itmay allow for clean separation of the frequency components. The absoluterotation rates should be chosen such that the lowest frequency componentin the signal falls above the 1/f noise corner of the sensor, within theapproximately white noise region. For many magnetometers (e.g. SQUIDs,fluxgates, AMR sensors) this corner frequency is in the order of 1 Hz.The minimum rotation rate should also be high enough that the relativechanges in measured fields are small during a single complete cycle ofthe orbital motion. In some applications, where the relative velocity ofthe source and sensor may be high, the optimal rotation rate should behigh enough to ensure that the measured field does not changesubstantially during one orbit.

The maximum lengths of the baselines a and b may be dictated by theavailable space within the instrument housing, which depends on theapplication. For downhole deployment, for example, baselines may berestricted to a few centimeters. Within the restrictions imposed byavailable space, it may be desirable for the baselines a and b to be ofsimilar length, so that the resolution and noise levels of differentgradient components are comparable.

The mechanism 120 may be configured to cause the sensor 110 to moverelative to the fixed instrument reference frame X in a predeterminedtrajectory as described in relation to any one of FIGS. 2A to 10D, orany other described trajectories. In trajectories comprising acombination of two rotations having angular velocities ω₁ and ω₂, thefirst rotation motion ω₁ may be driven by a first actuator 150, and thesecond rotation motion ω₂ may be driven by a second actuator 150.Alternatively, the mechanism 120 may be configured to cause both thefirst and second rotation motions and be driven by a single actuator150. For example, the mechanism 120 may include a set of cogs or gearswith a predetermined ratio to couple the first rotation motion to thesecond rotation motion with a predetermined ratio between ω₁ and ω₂ anda fixed relative phase.

Referring to FIG. 11, a schematic diagram of a measuring instrument 100is shown according to some embodiments. The measuring instrument 100comprises: a sensor 110, a mechanism 120, a signal processor 130, apower source 140, actuators in the form of first and second motors 152and 154, first and second angular position sensors 182 and 184respectively associated with motors 152, 154, a controller 160, and acomputer processor 170 as described in relation to FIG. 1.

The measuring instrument 100 comprises a body 102 which is fixedrelative to a fixed reference frame X(x, y, z) of the instrument 100.The mechanism 120 is configured to cause relative motion between thesensor 110 and the fixed reference frame X of the instrument 100.

The mechanism 120 comprises a first member 1110 having a first axis 1112and a second axis 1114 that is different from the first axis 1112. Themechanism 120 causes the first member 1110 to rotate about the firstaxis 1112, and causes the sensor 110 to rotate about the second axis1114. The sensor 110 is spatially offset from the first axis 1112. Insome embodiments, the sensor 110 is also spatially offset from thesecond axis 1114. The first axis 1112 may be spatially offset from thesecond axis 1114.

The mechanism 120 comprises a first sub-mechanism 1120 to cause thefirst member 1110 to rotate about the first axis 1112 and a secondsub-mechanism 1140 to cause the sensor 110 to rotate about the secondaxis 1114. The second sub-mechanism 1140 may comprise the first member1110. The first sub-mechanism 1120 may comprise a support 1130 tosupport the first member 1110. The support 1130 may be in the form of asleeve, for example.

The first sub-mechanism 1120 is driven by the first motor 152 via afirst drive shaft 1122, a first gear 1124 connected to the first driveshaft 1122, and a second gear 1126 coupled to the first gear 1124 andconnected to the support 1130. The first and second gears 1124, 1126 maybe in the form of spur gears. The first angular sensor 182 may beconnected to the signal processor 130 and configured to measure anangular position of the first drive shaft 1122 to determine an angularposition of the first sub-mechanism 1120.

The second sub-mechanism 1140 is driven by the second motor 154 via asecond drive shaft 1142, a third gear 1144 connected to the second driveshaft 1142, and a fourth gear 1146 coupled to the third gear 1144 andconnected to the first member 1110. The third and fourth gears 1144,1146 may be in the form of bevel gears. The second angular sensor 184may be connected to the signal processor 130 and configured to measurean angular position of the second drive shaft 1142 to determine anangular position of the second sub-mechanism 1140.

The first member 1110 may comprise an elongate arm. The secondsub-mechanism 1140 may further comprise an arm extension 1116 tospatially offset the sensor 110 from the second axis 1114.

The first member 1110 may be supported in bearings 1138 mounted in thesupport 1130. The first member 1110 may further comprise radialprotrusions or flanges 1118 configured to abut the bearings 1138 torestrict axial movement of the first member 1110 relative to the support1130. The second drive shaft 1142 may also comprise radial protrusions1148 configured to abut further bearings 1138 mounted in the support1130 to restrict axial movement of the support 1130 relative to thesecond drive shaft 1142. Further bearings 108 may be mounted in theinstrument body 102 and the support 1130 may be mounted in the bearings108. The first and second motors 152, 154 may be mounted in theinstrument body 102 and fixed with respect to the fixed reference frameX of the instrument 100.

The sensor 110 may be connected to the signal processor 130 by a cable1160. The cable 1160 may extend through the arm extension 1116, thefirst member 1110 and the second drive shaft 1142 into the instrumentbody 102 to connect the sensor 110 to the signal processor 130. Thefirst angular position sensor 182 may be connected to the signalprocessor 130 by cable 1162, and the second angular position sensor 184may be connected to the signal processor 130 by cable 1164.

Some embodiments relate to a drilling system 1200 comprising a measuringinstrument 100 as shown in FIG. 12. The drilling system 1200 maycomprise a drilling rig 1205 to drive a drill string 1210 with a drillbit 1220. The drilling system 1200 may be used to cut into earth or rock1230 to form a wellbore, hole or borehole 1235. The drill string 1210may extend between the drilling rig 1205 and the drill bit 1220 withinthe borehole 1235 to allow the drilling rig 1205 to drive the drill bit1220 to cut into the earth 1230. The drill string 1210 may comprise ahollow drill pipe configured to deliver drilling mud to the drill bit1220. The drilling mud may be pumped from drilling rig 1205 at thesurface, through the drill string 1210 to the drill bit 1220 and thenflow back up the borehole 1235 through an annulus formed between thedrill string 1210 and walls of the borehole 1235. The measuringinstrument 100 may be mounted to the drill string 1210 and may bemounted relatively near to the drill bit 1220.

In some embodiments, a complete measuring instrument 100 as describedabove may be mounted on the drill string 1210. In such embodiments, thesignal measured by the sensor 110 may be stored on memory in thecomputer processor 170 and accessed later, after the instrument 100 hasbeen retrieved from the borehole 1235.

In other embodiments, some of the components of the measuring instrument100 may be mounted on the drill string 1210 (downhole components) andother components of the measuring instrument 100 may be disposed at aremote location from the downhole components (remote components). Someof the remote components may be disposed in a remote module 1250 whichmay be disposed at the surface near the drilling rig 1205, or at anotherremote location.

The remote components may include any one or more of: the power source140, signal processor 130, computer processor 170, controller 160, andthe user interface 175. The downhole components may include the sensor110 as well as any or all of the rest of the components of theinstrument 100. Although, if all of the components of the instrument aredownhole components mounted on the drill string 1210, then remotecomponents may not be required.

In some embodiments, the downhole components may include the sensor 110,mechanism 120, actuator 150, angular position sensor 180, controller160, power source 140, and a transmitter (not shown) to transmit themeasured signals to the remote module 1250 comprising a receiver toreceive the measured signals and input them to the signal processor 130coupled to the computer processor 170 and user interface 175. The signaltransmission may be performed wirelessly or via a cable or by mud-pulsetelemetry.

The actuator(s) 150, or one of the actuators 150 could also be a remotecomponent with a mechanism 120 capable of coupling the remote actuator150 to the sensor 110 to drive the motion of the sensor 110. Inembodiments with one or more remote actuators 150, the angular positionsensors 180 may be coupled to or disposed near the actuator 150. Forexample, in a drilling application, one or more actuators 150, angularposition sensors 180 or other components of the instrument 110 may belocated at the surface, while the sensor system comprises downholecomponents located near a drill bit of the drilling system.Alternatively, the angular position sensors 180 may be downholecomponents, coupled to the sensor 110 or to the mechanism 120 anddisposed near the sensor 110.

In some embodiments, the actuator 150 or one of the actuators 150 maycomprise the drilling rig 1205. In embodiments with simple sensortrajectories comprising a single component of rotation ω₁, the sensor110 may simply be fixedly mounted to the drill string 1210 at somedistance from an axis of rotation of the drill string 1210. The drillingrig 1205 and drill string would then comprise the actuator 150 andmechanism 120 and drive the orbital motion of the sensor 110.

In embodiments with sensor trajectories comprising a second component ofrotation ω₂, the instrument 100 may comprise a second actuator 150 suchas an electric motor mounted to the drill string 1210 to drive the ω₂motion. If an electric motor is used and the sensor 110 is amagnetometer, the magnetic field of the motor should be taken intoconsideration when configuring the instrument 100. Magnetic interferencefrom the motor may be mitigated by positioning the motor away from thesensor 110, using magnetic shielding to shield the motor (i.e., toshield the sensor 110 from the motor), and/or driving the rotation ofthe sensor 110 at frequencies that differ from the rotation rate of themotor so that interference from the motor can be identified and filteredout during signal processing.

Alternatively, the mechanism 120 may be configured to move the sensor110 through both primary and secondary rotations being driven by asingle actuator 150. For example, the rotation of the drilling rig 1205may drive both primary and secondary rotations to move the sensor 110through its trajectory without the need for a second actuator 150. Thismay be done through a set of gears, for example.

In some embodiments, the primary and/or secondary rotations may bedriven by a downhole turbine mounted to the drill string 1210 andpowered by the flow of drilling mud through the drill string 1210.Referring to FIGS. 13A to C, the downhole components of one suchembodiment are shown.

The downhole components of the measuring instrument 100 shown in FIGS.13A to 13C include a first sensor 110 (S1), a second sensor 112 (S2), amechanism 120, an angular position sensor 184, and an actuator in theform of a turbine 1350. The mechanism 120 is configured to move thesensors 110, 112 in the trajectory described in relation to FIGS. 4A to4D.

The mechanism 120 comprises a first member 1310 having a first axis 1312and a second axis 1314 that is different from the first axis 1312. Thefirst member is in the form of an elongate rod 1310 with the sensors110, 112 fixedly mounted at opposing ends of the rod 1310. The sensors110, 112 are mounted such that sensitive axes u₁ and u₂ of the sensors110, 112 respectively remain parallel to each other. The first axis 1312is the longitudinal axis of the drill string 1210 and the second axis1314 is the longitudinal axis of the rod 1310.

The angular position sensor 184 is coupled to the rod 1310 to measurethe angular position φ of the sensors 110, 112, and the signal measuredby the angular position sensor 184 is then transmitted to the signalprocessor 130 via cable 1364. The angular position sensor 184 may bedisposed within the lumen of the drill string 1210 as shown, or in someembodiments, may be disposed outside the drill string 1210 such as nearthe sensors 110, 112, for example. Another cable 1360 is coupled to thesensors 110, 112 to transmit the signal measured by the sensors 110, 112to the signal processor 130.

The mechanism 120 comprises a first sub-mechanism 1320 to cause the rod1310 to rotate about the first axis 1312 (i.e., the primary rotation ω₁)and a second sub-mechanism 1340 to cause the sensors 110, 112 to rotateabout the second axis 1314 (i.e., the secondary rotation ω₂). The firstsub-mechanism 1320 comprises the drill string 1210, which is driven byan actuator 150 which in this case comprises the drilling rig 1205.

The second sub-mechanism 1340 comprises the rod 1310 pivotally coupledto the drill string 1210 and is driven by the turbine 1350. The rod 1310may be mounted in sealed bearings (not shown) so that the rod 1310passes through the drill string 1210 from one side to another. Theturbine 1350 is mounted to the rod 1310 such that the turbine 1350 isdisposed within a lumen of the drill string 1210.

When drilling mud is pumped through the drill string 1210 in thedirection shown by arrow 1330, the turbine 1350 is driven by thepressure of the drilling mud to rotate the rod 1310 and consequently thesensors 110, 112 to cause the secondary rotation ω₁. Alternatively, theturbine 1350 could be disposed in the annulus formed between the drillstring 1210 and the walls of the borehole 1235 and driven by thepressure of the drilling mud returning back up the borehole 1235 to thesurface.

Together, the first and second sub-mechanisms cooperate to move thesensors 110, 112 in the trajectory shown in FIGS. 4A to 4D. Simplyremoving one of the sensors 110, 112 would result in the single sensortrajectory shown in FIGS. 3A to 3D.

Various other mechanisms 120 may be configured to cause one or moresensors 110 to move through any one of the sensor trajectories describedabove.

It will be appreciated by persons skilled in the art that numerousvariations and/or modifications may be made to the above-describedembodiments, without departing from the broad general scope of thepresent disclosure. Therefore, the present embodiments are, to beconsidered in all respects as illustrative and not restrictive.

1. A measuring instrument comprising: a sensor to measure a property ofthe local environment; a mechanism configured to cause the sensor tomove along a predetermined path relative to a fixed reference frame ofthe instrument; and a signal processing system configured to receive asensor signal generated by the sensor, perform a Fourier transform onthe sensor signal to identify frequency components of the sensor signal,and compare the frequency components of the sensor signal with frequencycomponents associated with the predetermined path to determine ameasurement of the property of the local environment; wherein themechanism comprises a first member, wherein the first member has a firstaxis and a second axis that is different from the first axis, whereinthe mechanism is configured to cause the first member and the sensor torotate about the first axis, and to cause the sensor to rotate about thesecond axis, wherein the sensor is spatially offset from the first axis,wherein rotation of the sensor and the first member about the first axishas a first angular velocity and rotation of the sensor about the secondaxis has a second angular velocity, and wherein rotation of the sensorand the first member about the first axis is coupled to rotation of thesensor about the second axis such that the first angular velocity isrelated to the second angular velocity by a predetermined ratio betweenthe first and second angular velocities.
 2. A measuring instrumentaccording to claim 1, (i) wherein the sensor is spatially offset fromthe second axis, (ii) wherein the second axis is spatially offset fromthe first axis, or (iii) wherein the sensor is spatially offset from thesecond axis and wherein the second axis is spatially offset from thefirst axis.
 3. (canceled)
 4. A measuring instrument according to claim21, wherein the second axis is perpendicular to the first axis.
 5. Ameasuring instrument according to claim 21, wherein the second axis isinclined at an acute angle relative to the first axis.
 6. A measuringinstrument according to claim 21, wherein the first and second angularvelocities are phase-locked.
 7. A measuring instrument according toclaim 21, wherein the first and second angular velocities are variablewith time.
 8. A measuring instrument according to claim 1, wherein themechanism comprises: a first sub-mechanism to cause the first member andthe sensor to rotate about the first axis; and a second sub-mechanism tocause the sensor to rotate about the second axis, wherein the secondsub-mechanism comprises the first member, and wherein the firstsub-mechanism comprises a support to support the first member. 9-20.(canceled)
 21. A measuring instrument according to claim 1, wherein thesensor is a first sensor and the measuring instrument further comprisesa second sensor offset from the first axis on an opposite side of thefirst axis from the first sensor, and wherein the mechanism isconfigured to cause the second sensor to rotate about the first axis inthe same direction and with the same angular velocity as the firstsensor, and to cause the second sensor to rotate about the second axisin the same direction and with the same angular velocity as the firstsensor, and wherein the signal processing system is configured toreceive sensor signals generated by the first and second sensors,perform Fourier transforms on the sensor signals to identify frequencycomponents of the sensor signals, and compare the frequency componentsof the sensor signals with frequency components associated with thepredetermined path to determine a measurement of the property of thelocal environment.
 22. A measuring instrument according to claim 21,wherein rotation of the second sensor about the second axis is out ofphase relative to rotation of the first sensor about the second axis.23. A measuring instrument according to claim 21, wherein rotation ofthe second sensor about the second axis is about ninety degrees (90°)out of phase relative to rotation of the first sensor about the secondaxis.
 24. A measuring instrument according to claim 1, furthercomprising a second member coupled to the first member, the secondmember having a third axis which is different from the first and secondaxes, wherein the mechanism is configured to cause the second member torotate about the first axis. 25-44. (canceled)
 45. A measuringinstrument according to claim 21, (i) wherein the sensors are configuredto measure the field strength, (ii) wherein the sensors are configuredto measure one or more vector components of a local force field, or(iii) wherein the sensors are configured to measure the field strengthand wherein the sensors are configured to measure one or more vectorcomponents of a local force field.
 46. A measuring instrument accordingto claim 45, wherein the local force field to be measured is one of: amagnetic field, an electric field, and a gravitational field.
 47. Ameasuring instrument according to claim 21, wherein the sensors compriseone or more selected from: a magnetometer, a total magnetic intensitymagnetometer, a uniaxial magnetometer, and a fluxgate magnetometer.48-55. (canceled)
 56. A measuring instrument according to claim 21,further comprising one or more angular position sensors connected to thesignal processor to provide angular position information for one or morecomponents of the mechanism. 57-71. (canceled)
 72. A drilling systemcomprising: a measuring instrument according to claim 8; a drill bit forcutting a borehole in rock or earth; a drill string to drive the drillbit, the drill string having a lumen to deliver drilling mud to thedrill bit; and a drilling rig to drive rotation of the drill string anddrill bit, wherein at least part of the measuring instrument is mountedto the drill string. 73-74. (canceled)
 75. A drilling system accordingto claim 72, wherein at least part of the mechanism is driven by thedrilling rig.
 76. A drilling system according to claim 72, wherein atleast part of the mechanism is driven by drilling mud pressure via aturbine coupled to the mechanism.
 77. A drilling system according toclaim 72, wherein the first sub-mechanism comprises the drill string,and wherein the second sub-mechanism comprises a turbine positionedwithin the lumen of the drill string to drive the second sub-mechanism.78. (canceled)
 79. A drilling system comprising: a measuring instrumentaccording to claim 21; a drill bit for cutting a borehole in rock orearth; a drill string to drive the drill bit, the drill string having alumen to deliver drilling mud to the drill bit; and a drilling rig todrive rotation of the drill string and drill bit, wherein at least partof the measuring instrument is mounted to the drill string.